Compensation for non-linear distortion in a modem receiver

ABSTRACT

A method and modem receiver for receiving serial data from a transmission link, for recovering data from the received data, and for synchronizing the receiver timing to that of the transmitter are disclosed. A receiver equalizer includes a filter and quantizer in the forward path and a feedback circuit connecting the quantizer output to the filter input. A measuring circuit is used to initialize the feedback circuit. The receiver filter, provided as a QMF wavelet filter bank, is implemented in one embodiment with a form of polyphase filtering followed by four M/2-point FFTs. The quantizer recovers data symbols and provides a quantization error feedback signal. The measuring circuit provides an approximation of phase and amplitude of isolated wavelets.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 09/943,697 filed Aug. 31, 2001, now U.S. Pat. No. 6,735,264,entitled COMPENSATION FOR A NON-LINEAR DISTORTION IN A MODEM RECEIVER.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

N/A

BACKGROUND OF THE INVENTION

As is known in the art, a modulator-demodulator (modem) is an electronicdevice that modulates transmitted signals and demodulates receivedsignals. The modem generally provides an interface between digitaldevices and an analog communications system to thus make possible analogtransmission of digital information between two terminals or stations.Such transmissions may be over a transmission link such as a telephoneline, cellular communication link, satellite link, and cable TV, each ofwhich being generally band-limited. That is, the information may betransmitted across the transmission link only over a predetermined rangeof frequencies having a maximum bit error rate.

As is also known, a modem is used to provide wireless transmissionbetween transmitting and receiving stations. Such wireless communicationcan be employed in a variety of applications: VHF, IS-54 (cellular),IS-95 (cellular), SPADE (satellite), GSM (cellular), HDTV, and SAT-TV,each using one of the following linear modulation techniques: QAM; QPSK;Pi/4 DPSK; GMSK. As with the transmission line application, thebandwidth of each is limited within an acceptable bit error rate.

While most modems are capable of providing compensation for Guassiannoise, impulse noise is not well managed. Most modems also requirehigher powered amplification means since, in all cases, amplitudedistortion is unacceptable. In broadcast environments, it is wellunderstood that FM transmission provides superior impulse noisehandling. Also in FM transmission schemes, the transmitted signal may beamplified with almost 100% efficiency since the information carryingportion of the signal is identified by the zero-crossings of the signal.Thus, amplitude distortion is ignored.

Despite these apparent advantages, only a few isolated examples exist ofnonlinear frequency modulation used within modems. Frequency shiftkeying, providing a 300 bits per second data rate, essentially utilizedtwo frequencies, each for representing a separate data bit. The nextmajor development in modems was the use of phase modulation, beginningwith two phase modulation, then four, then eight. A combination ofamplitude and phase modulation was later developed, also referred to asquadrature amplitude modulation or QAM.

A subsequent development was Guassian minimum shift keying, or GMSK. Atfirst glance, such coding resembles four phase modulation, though inorder to avoid amplitude modulation, a special low pass filter referredto as a Guassian filter was applied to the data going into the phasemodulator. Since some have regarded the Guassian filter as akin to anintegrator, an argument could be made that such a modulation schemeresults in frequency modulation, since GMSK can be demodulated with anFM discriminator. Yet, GMSK applies linear functions of the data toin-phase and quadrature carriers to produce a linear modulation;whereas, true FM is mathematically equivalent to the application oftrigonometric functions of the filtered signal to an in-phase andquadrature carrier. FM is a non-linear modulation, which was perceivedas inefficient for data transmission since the transmitted frequencyspectrum is not a simple translation of the baseband spectrum as in AMmodems. A further perceived problem with FM for a modem is that it onlyaccepts real signals into its voltage controlled oscillator. That is, inthe equivalent in-phase and quadrature carrier method for FM, both pathssend the same data resulting in a double sideband spectrum of a singlecarrier, which was regarded as redundant and therefore inefficientcompared to the double-sideband spectra of the dual in-phase andquadrature carriers of QAM modems. Also, single-sideband (SSB)transmission was considered undesirable for modems because there is nosimple way to efficiently demodulate an AM-SSB signal without a carrierreference. FM was dismissed as an analog technique totally unrelated todata transfer except with regard to FSK.

In an ordinary modem, binary data is normally passed through a basebandraised-cosine filter which limits the bandwidth of the baseband signalso that when one multiplies the baseband signal by a carrier, controlover the passband signal bandwidth is provided without intersymbolinterference. The output of an ordinary modem includes signals havingdiscrete phases such that data included therein can be identified bydiscerning the phase of each bit. For instance, whenever a signal has a+90° phase shift it is interpreted as 0 and when the phase shift is −90°it represents 1, etc. Thus, in an ordinary modem employing a carrier,the phase and/or amplitude of the carrier signal are determined by thecurrent symbol being transmitted. The carrier assumes only selectedvalues of phase and amplitude for most of the duration of each symboland graphical plots of all the selected phase-amplitude pairs are calledthe constellation points of the modem. Ordinary modems require thatthere be distinct points in the constellation for each possible value ofthe transmitted symbol. Furthermore, bit errors occur in the receiver ifthe points are mis-assigned due to intersymbol interference or due tonoise on the link.

In many applications the computational requirements of the modemintroduce a delay which is detrimental to the operation of the system.For example, digital voice transmission and multiple access networks aresensitive to delay in the modem. Furthermore, the rate at which a modemmay transmit and receive data per unit of bandwidth is called the modembandwidth efficiency. In the discipline of Digital Information Theorythis efficiency is known to be maximized when the transmitted signal hasthe maximum entropy or randomness. The maximum entropy transmission isband-limited Guassian noise, and among other properties Guassian noisewill not dwell at a distinct phase-amplitude pair as in a constellation.Thus it is desirable to provide a modem having a passband carrier whichminimizes the internal processing time and also maximizes the bandwidthefficiency without sacrificing bit error rate.

A method of communicating a sequential series of symbols over atransmission link comprises the steps of multi-rate polyphase filteringthe symbols, using the filtered output to modulate a carrier,transmitting the modulated carrier across the transmission link,receiving the modulated signal, applying the inverse of the transmitterpolyphase filter to the received signal, and thresholding andre-assembling the output of the inverse filter to recover thetransmitted symbols.

Data is transmitted from a first modem to a second modem across awireless transmission link by forming input data frames from the inputdata, multiplying the input frames by a rotation matrix, frequencymodulating and transmitting the rotation matrix output, receiving andfrequency demodulating the transmitted data, multiplying the demodulatedsignal by a second rotation matrix, and re-assembling the de-rotateddata to recover the original data.

A modem for communicating symbols across a transmission link includes atransmit portion and a receive portion, wherein the transmit portioncomprises a partitioning element for dividing the input into paralleldata channels, a baseband transmit rotation section for polyphasetransforming the channeled data into parallel signal channels, are-arrangement of the parallel signal channels into sequential serialsamples, a carrier modulator for providing a modulated signal, and atransmitter for transmitting the modulated signal. The receive portioncomprises a receiver for receiving the transmitted, modulated signal, ademodulator for demodulating the received signal into parallel signalchannels, a receive rotation section for polyphase transforming thedemodulated, received signal into parallel data channels, and anassembling element for combining the parallel data channels into aserial data signal.

The polyphase filtration of these methods and this modem enables FMmodulation of the transmitted symbols since only a real component of theoriginal symbols is generated. FM, whether achieved using frequencymodulation or phase modulation of the input signal, provides enhancedimmunity to non-Guassian noise, provides high bandwidth efficiency,utilizes non-coherent IF and thus requires no carrier recovery, is oflower cost than conventional modems in part due to the absence of A/Dconverters, provides low co-interference due to the FM capture effect,and is compatible with analog signals, owing to the use of commutingoperators. More power efficient, but potentially less linear amplifierssuch as Class B and Class C can be employed since zero crossings areused to determine data content; carrier recovery is not required. Modemsemploying modulation schemes such as QAM cannot employ such non-linearamplification. Further, satellite modems employing the presentlydisclosed method and modem save on TWT backoff power and thus are moreenergy efficient since intermodulation is not a problem, while personalcomputers achieve higher data rates without sacrificing bit error rate.

Polyphase filtration is implemented in a first embodiment by a waveletfilter (e.g. Quadrature Mirror Filter) pair. The partitioning elementpartitions the serial data among plural, parallel data channels prior tolinear-phase FIR vector filtration, the filter coefficients being squarematrices, whereby the input data are transformed into parallel signalchannels. The transformation is by way of a convolutional rotation ofthe input data vector. Each coordinate of the output signal is confinedto a frequency sub-band which slightly overlaps its neighbor.Pre-emphasis in the transmit portion, prior to the rotation in a firstembodiment, places most of the information in the lower basebandfrequencies. This is due to the noise probability density function of anFM discriminator that is proportional to the square of the frequency.De-emphasis in the receiver results in an addition to the overall gainequation. This equation, in one embodiment, includes contributions bythe FM transmitter gain, the de-emphasis gain, and a noise reductiongain. The pulse amplitude levels representing the partitioned data bitswithin each sub-band need not necessarily correspond to an integernumber of bits, as long as all of the levels in all the subbandscorrespond to an integer value.

The receiver portion provides the de-rotation filter for performing areverse transformation to recover the original data. In one embodiment,the reverse transformation commutes with the modulation transformation.In a further embodiment, the coefficients of the de-rotation filter inthe receiver are adaptively selected for equalization to correct fortransmission path distortion, since the analyzer is a fractional-rateFIR filter. Thus, a near-perfect reconstruction filter is employed. Athreshold operator takes the nearest integer coordinate values as themost likely symbol.

In a further embodiment, the commuting rotation and de-rotation filtersare derived from the elementary matrices which describe a geometricrotation of a vector. Their function is to transform an input datavector within a data coordinate system into a signal vector within asignal coordinate system such that the sequentially serializedcoordinates of the signal vector would form the digital samples of abandlimited analog signal. Yet another approach to the matrices ismathematical transformation by way of discrete wavelet transformations.

In each of the transmitter portion and receiver portion, the rotationoperator is ideally a computationally efficient multi-rate waveletfilter bank. Logarithmic amplification of the baseband signal prior tointroduction into the FM transmitter modulator results in an improvementin the modulation gain out of the receiver. Further, as a by-product ofthe logarithmic amplification prior to the transmitter andde-amplification after the receiver, noise introduced in the transmitchannel is attenuated.

Dispersive impairments in the link may create a relative phase shiftbetween sub-bands. In that case, the “orthogonality” of the wavelets islost and cross terms appear as self-interference in the recoveredsymbols.

BRIEF SUMMARY OF THE INVENTION

In the presently disclosed invention, a modem receiver enablesequalization and synchronization to the respective transmitter. Amultiband receiver equalizer uses data in each sub-band from a QMFfilter bank to continuously track the phase and amplitude of eachsub-band without the use of pilot tones. The modem uses this informationfor both data recovery by the equalizer and as the phase detector inputto a phase-locked loop used for transmitter timing synchronization.

The equalizer is comprised of a filter and quantizer in the forwardpath, a feedback circuit connecting the quantizer output to the filterinput, and a measuring circuit used to initialize the feedback circuit.

The receiver multirate QMF filter bank may be implemented with a form ofpolyphase filtering followed by four M/2-point FFTs. The FFT outputssuccessive sub-bands, one per sample. The receiver filter is thencapable of removing the effects of cross-correlations resulting from thetransmitter, the transmission medium, or both.

Multipath echo compensation using the disclosed receiver circuitry isalso discussed, as is the derivation of preferred wavelet functionsutilized in the presently disclosed modem.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

The foregoing features of this invention as well as the invention itselfmay be more fully understood from the following detailed description ofthe drawings in which:

FIG. 1 is a diagrammatical representation of a signaldecomposition-recomposition system;

FIG. 1A is a diagrammatical representation of a cascade analyzer;

FIG. 1B is a diagrammatical representation of a cascade synthesizer;

FIG. 1C is a diagrammatical representation of a tree analyzer;

FIG. 1D is a diagrammatical representation of a tree synthesizer;

FIG. 2 is a block diagram of a signal encryptor;

FIG. 3 is a block diagram of a transmitting and receiving system for thetransmission and reception of secure signals;

FIG. 4 is a block diagram of a signal encrypting transmitting andreceiving system;

FIG. 5 is a block diagram of a modem;

FIG. 6 is a block diagram of a coded modem;

FIG. 7 is a block diagram of a coded tree modem;

FIG. 8 is a block diagram of a system for receiving and transmittingcompressed signals;

FIG. 9 is a block diagram of a signal compression system fortransmitting and receiving signals over a digital link;

FIG. 10 is a block diagram of a modem;

FIG. 11 is a block diagram of an FM modem according to the presentinvention;

FIG. 12 is a flow diagram of a method for transmitting data using adouble-side band variant of the FM modem of FIG. 11;

FIG. 13 is a flow diagram of a method for transmitting data using asingle-side band variant of the FM modem of FIG. 11;

FIG. 14 is a flow diagram of sub-steps invoked by the method of FIGS. 12and 13;

FIG. 15 is a flow diagram of a method for receiving data using the FMmodem of FIG. 11;

FIG. 16 is a flow diagram of sub-steps invoked by the method of FIG. 15;

FIG. 17 is a block diagram of a modem receiver according to thepresently disclosed invention;

FIG. 18 is an exploded block diagram of the modem receiver of FIG. 17;

FIG. 19 is a block diagram of an Equalizer as used in the modem receiverof FIG. 17;

FIG. 20 is a block diagram of a Phase Measurement Unit for use in theEqualizer of FIG. 19;

FIG. 21 is a block diagram of a Phase Measurement Loop for use in theEqualizer of FIG. 19;

FIGS. 22A and 22B are graphs illustrating proportional phase error forselected symbol values;

FIG. 23 is a block diagram of an Amplitude Measurement Loop for use inthe Equalizer of FIG. 19;

FIG. 24 is a block diagram of an Amplitude Measurement Unit for use inthe Equalizer of FIG. 19;

FIG. 25 is a block diagram of timing and phase synchronization circuitryfor the modem receiver of FIG. 17;

FIG. 26 is a block diagram of a Preamble Detect circuit as used in thetiming and phase synchronization circuitry of FIG. 25;

FIG. 27 is a block diagram of a Sync Predict circuit as used in thetiming and phase synchronization circuitry of FIG. 25;

FIG. 28 is a block diagram of a Sync Detect circuit as used in thetiming and phase synchronization circuitry of FIG. 25;

FIG. 29 is a block diagram of the Sync Detect circuit of FIG. 28 and aSlide Balance circuit as used in the timing and phase synchronizationcircuitry of FIG. 25;

FIG. 30 is a block diagram of a Capture Frequency circuit for use in thereceiver of FIG. 17;

FIG. 31 is a block diagram of a Frequency Lock circuit for use in thereceiver of FIG. 17;

FIG. 32 is a block diagram of algebra utilized in the definition ofwavelet functions used to transmit data to the modem receiver of FIG.17;

FIG. 33 is a block diagram of a process utilized in the definition of atransmitter signal for receipt at the modem receiver of FIG. 17;

FIG. 34 is a phase diagram useful in characterizing a multipath problemaddressable by the modem receiver of FIG. 17;

FIG. 35 is a block diagram of a technique used in the modem receiver ofFIG. 17 to address multipath effects; and

FIG. 36 is a block diagram of a filter circuit as used in the Equalizerof FIG. 18.

DETAILED DESCRIPTION OF THE INVENTION

Referring now to FIG. 1 a signal decomposition-recomposition system 10includes an analyzer 12 having an input port 14 a and a pair of outputports 16 a, 16 b. Each of the output ports 16 a, 16 b are coupled to acorresponding pair of input ports 18 a, 18 b of a synthesizer 20.

An analog input signal X fed to the analyzer input port 14 a isdecomposed into a pair of signals W′ and V′ each of which is fed to acorresponding one of the output ports 16 a, 16 b. Similarly, a pair ofinput signals W′, V′ fed to the synthesizer input ports 18 a, 18 b arereconstructed in to an output signal Y at a synthesizer output port 22a. The decomposition and reconstruction processes performed by theanalyzer 12 and the synthesizer 20 as well as the signals V′ and W′ willbe described further hereinbelow. Suffice it here to say that the inputsignal X is decomposed into the signals V′ and W′ such that the signalsV′ and W′ may later be combined to exactly reconstruct the input signalX.

It should be noted that the sub-analyzer and sub-synthesizer buildingblocks operate digitally, however, for clarity in the description, thesignal conditioning circuits required to convert between analog signalsand digital signals in the system have been omitted. Such signalconditioning circuits will be described in conjunction with FIG. 4below. Briefly, however, and as will be described in conjunction withFIG. 4, when an input signal to a system building block, e.g. asub-analyzer or sub-synthesizer, corresponds to an analog signal, thesignal should first be fed through a filter having filtercharacteristics selected to provide Nyquist filtering. The appropriatelyfiltered signal may then be sampled by an analog-to-digital converter(ADC). Similarly, if the output from a system building block is to be ananalog signal, the signal should be fed to a digital-to-analog converter(DAC) and fed to a second filter likewise selected having filtercharacteristics selected to provide Nyquist filtering.

The sub-synthesizer 20 performs the inverse operation of thesub-analyzer 12, and vice versa. That is, if the signals W′ and V′ froma sub-analyzer are applied as input signals to the sub-synthesizer inputports 18 a, 18 b, the output signal Y will be identical to the sequenceof input samples, X, except for a predetermined time delay. In apreferred embodiment, the time delay corresponds to one sample time.

Likewise, applying the signals V′, W′ to the respective synthesizerinput ports 18 a, 18 b and then applying the resultant output signal Yto the analyzer input port 14 a should provide at the respectiveanalyzer output ports 16 a, 16 b the original input signals V′ and W′.

As described above, the analyzer 12 and synthesizer 20 provide signaldecomposition and reconstruction functions. As will be described below,the sub-analyzer and sub-synthesizer may be used as building blocks andmay be coupled to provide more complex circuits which may themselves becoupled to provide a variety of signal transmitting and receivingsystems. Furthermore, the analyzer 12 and synthesizer 20 as well asother system building blocks to be described hereinbelow may beefficiently implemented in hardware, software or by a combination ofhardware and software.

The analyzers and synthesizers to be described herein operate on anordered sequence of numbers, which may be, but are not limited to,samples from an analog to digital converter (ADC). For example, samplesmay be expressed as X(0), X(1), X(2), and X(3) where X(0) is the mostrecent sample. Each of the binary numbers X(0) through X(3) has aparticular value within a predetermined range of values. An 8-bit ADC,for example, may provide a range of decimal values between −128 to +127.

The sequence of samples X(0) through X(3) may be considered as thecoordinates of a vector [X] in a 4-dimensional space. A lineartransformation may be made from the coordinate system of X to anothercoordinate system in the same 4-dimensional space. Thus, the vector Xmay be transformed to a vector Y by a “rotation” matrix C. In matrixnotation this may be expressed as:[Y]=[C]*[X]

Using axes of the new coordinate system that are mutually orthogonal,the vector Y has components which correspond to the projections of Xonto the new transformation axes. Such projections are provided byforming a vector dot product.

In a 4 dimension case for example, a set of Walsh codes, Code 1–Code 4,may be provided as:

-   -   Code 1=+1+1+1+1    -   Code 2=+1−1+1−1    -   Code 3=+1+1−1−1    -   Code 4=+1−1−1+1        The Walsh codes Code 1–Code 4 above represent an orthogonal        coordinate axis in the 4-dimensions of this time ordered space.        Codes 1–4 have a length corresponding to the square root of 4,        (i.e. the dot product of code 1 with itself equals 4) and thus        are not unit vectors.

The rotation matrix C may be expressed as:

$\begin{matrix}\; & {C_{1} = {{+ 1} + 1 + 1 + 1}} \\\; & {{\lbrack C\rbrack = {C_{2} = {{+ 1} - 1 + 1 - 1}}}\mspace{65mu}} \\\; & {C_{3} = {{+ 1} + 1 - 1 - 1}} \\\; & {C_{4} = {{+ 1} - 1 - 1 + 1}}\end{matrix}$

That is, the rows C₁–C₄ of the rotation matrix C correspond to thecomponents of the Walsh code vectors such that matrix multiplicationbetween the rotation matrix C and the vector X is equivalent to the dotproduct of a particular one of the vectors C₁–C₄ with the vector X. Thecomponents of the row vector X may be expressed as:[X]=[X(0), X(1), X(2), X(3)]A new state vector Y may be expressed as:[Y]=[[C ₁ ]*[X], [C ₂ ]*[X], [C ₃ ]*[X], [C ₄ ]*[X]]where * symbolizes a dot product operation between [X], and the Walshvectors [C₁] through [C₄]. The new state vector [Y] completely describesthe original state defined by [X] and may be computed on every group of4 samples X(0)–X(3).

The linear rotation operation is exactly invertible, and thus:[X]=[C^][Y]where [C^] is the inverse of [C]. The matrix C^ may be found from therelation:[C^]=1/L [C]where L corresponds to the dimension of a row or column vector of thematrix C^.

In a two dimensional case, Walsh vectors may be expressed as C₁=[+1,+1]and C₂=[+1,−1]. Thus, analyzers may be defined having 2 outputs usingthe 2-dimensional vector above.

Walsh vectors of any dimension may be generated by the two dimensionalvectors C₁, C₂. That is, substitution of a set of Walsh codes into a2-dimensional generator matrix provides a new Walsh code having twicethe dimension. Using this procedure an N-dimensional transformationneeded to provide an analyzer, may be provided. Thus, because of the waythat Walsh vectors are constructed from their two dimensional Walshgenerators, higher dimension analyzers and synthesizers may beconstructed from the 2-dimensional case.

With this matrix transformation method, matrix equations for providing amodem or a signal encryptor may be generated. The fundamental operationin the requisite computations is addition and subtraction of terms.

It should be noted that in the case of a cascade analyzer or synthesizerprovided using the matrix method with a matrix having N dimensions, thelowest frequency channel is operated on by the matrix vector C₁ and eachsubsequent channel i is operated on by a corresponding matrix vectorC_(i) until the highest frequency channel is operated on by the matrixvector C_(N). Thus, in the case of a 4 dimension matrix the lowestfrequency channel is operated on by the matrix vector C₁ and the highestfrequency channel is operated on by the matrix vector C₄. The case of atree analyzer or synthesizer provided by the matrix method will bedescribed in conjunction with FIGS. 1C and 1D below.

In another preferred method the equations which describe thesub-analyzer output signals V′ and W′ may be provided as:V′=SHIFT*X(n−1)−BN*W′  Equation 1W′=2*X(n−1)−X(n)−X(n−2)  Equation 2In which:

V′ corresponds to a scaling or filtering function of the sampled inputsignal X;

W′ corresponds to a residual or derivative of the sampled input signalX;

X(n) corresponds to the most recent input sample;

X(n−1) and X(n−2) correspond to the two previous input samples;

SHIFT corresponds to a variable set equal to a positive power of 2 (suchas 32, or 64, or . . . ); and

BN corresponds to a positive integer having a value between 0 and SHIFTand is preferred to have a relatively small value.

As will be explained further below the scaling function V′ and theresidual W′ defined by Equations 1 and 2 may be evaluated on alternateinput samples of the input signal X, such that the value of the residualW′ corresponds to twice the difference between the “center” sample,X(n−1), and the midpoint of a line connecting the two nearest neighborsof X(n−1) here, the two nearest neighbors being represented as X(n) andX(n−2). When Equations 1 and 2 are evaluated on alternate input samples,only the odd numbered (or alternatively only the even numbered) “centerpoints” are selected for the calculation. Thus, the output rate of thesub-analyzer corresponds to one-half the input rate of the input signalX.

The residual W′ may also be interpreted as a deceleration about thecenter point. Thus it is possible to define the residual W′ as anacceleration and substitute −W′ for W′ in the Equations 1 and 2 above toprovide an alternate and essentially equivalent expression for theresidual W′.

Other implementations using more nearest neighbor points and whichdefine the residual W′ as a higher order or first order derivative ofthe function evaluated about the center point, may also be used.

The interpretation of the residual W′(N) described above indicates thatthe residual W′(N) responds only to changes in the slope aroundalternate samples. Thus one characteristic of the residual W′ in thepresent invention is that the residual W′ is provided having a value ofzero when the slope about the center point is constant.

The conventionally defined residual W(n) is related to the effectiveresidual W′(n) as defined in the present invention byW(n)=B*W′(n)

As a consequence of the definition for the residual W′(n) in the presentinvention, signal compression based on linear or non-linear quantizationof the residual W′(n) (rather than the residual W(n) as defined in theconventional approach) may result in improved performance overcompression schemes based upon conventional definitions of the residualW(N) such as:W(N)=X(n−1)−V(N)

The sub-synthesizer operation may be described by equations 3 and 4below:Y(n−1)=Y′(n−1)/SHIFT  Equation 3Y(n)=Y′(n)/SHIFT  Equation 4in which:

SHIFT corresponds to a power of 2 andY′(n−1)=V′+BN*W′;Y′(n)=2*Y′(1)−Y′(−)−SHIFT*W′; and

Y′(−) corresponds to a saved, previously calculated (i.e. recursive),value of Y′(n).

By defining the variable SHIFT, as above and taking advantage of thefact that multiplication and division by a power of two is equivalent toa right or left shift on a binary computer, practical implementations ofthese equations may be provided with relatively little and simplehardware and are therefore preferred.

Thus in the approach of the present invention, the residual W′ has beendefined to provide a particular characteristic and the sequence V′ isselected to provide the remainder of the sequence.

Referring now to FIG. 1A, a so-called “cascade analyzer” 24 includes aplurality of here N sub-analyzers. The cascade analyzer 24 may beprovided by feeding the signal V′₁ from an output port of a firstsub-analyzer 24 a into the input port of a second sub-analyzer 24 b andfeeding a signal V′₂ from the output port of the second sub-analyzerinto the input port of a third analyzer (not shown) and so on. Theprocess continuing until a sub-analyzer 24N provides a signal V′_(N)having a predetermined sample rate.

For example, in a communication system it may be desirable to couple aplurality of say N sub-analyzers as described above where N is selectedsuch that a signal V′_(N) is selected such that the signal V′_(N) isprovided having a sample rate which is below twice the lower frequencycutoff of the communications link. Since each sub-analyzer halves thesample rate to its input, the various output signals, W′₁, W′₂ . . .W′_(N) of the cascade analyzer are provided at differing rates.

Referring now to FIG. 1B a so-called “cascade synthesizer” 26 may beprovided from a plurality of sub-synthesizer 26 a–26N which may be ofthe type described above in conjunction with FIG. 1. The cascadesynthesizer operates to provide the inverse operation of the cascadeanalyzer.

Referring now to FIG. 1C a so-called “tree analyzer” 28 may be providedby appropriately coupling a plurality of sub-analyzers 29 a–29 g each ofwhich may be of the type described above in conjunction with FIG. 1.Thus, in addition to a cascade emanating from the scale function V′(N),it is also possible to have residual cascades emanating from one or moreresidual sequences W′(N). That is, the residual sequences W′(N) maythemselves be considered as inputs for a multi-resolution analysis. Itis therefore, in that case, possible to provide both the scalingfunction sequence V′ and the residual sequence W′ having equal samesample rates. Here, the sample rate corresponding to one-eighth theoriginal sample rate.

The tree analyzer 28 may be provided by analyzing each of the residualsequence outputs W′ until all such sequences are brought down to onerate, which is normally below the Nyquist rate of the lower cutofffrequency of a transmission link with which the analyzer cooperates. Thethree level analyzer tree 28 is here provided having all of its output29 a–29 h at ⅛ the sample rate, r, of the analog input signal, X. Eachoutput sample may include many more bits per sample than the inputsample, however, the numbers represented by those output samples willusually be small in magnitude and easily re-quantized by a quantizer(not shown).

Referring now to FIG. 1D a so-called “tree synthesizer” 30 may beprovided from a plurality of sub-synthesizers 31 a–31 g, each of whichmay be of the type described above in conjunction with FIG. 1. The treesynthesizer 30 provides the inverse operation of the tree analyzer 28(FIG. 1C).

It should be noted that in each of the applications to be describedherein below in conjunction with FIGS. 2–8, the term analyzer as usedthroughout FIGS. 2–8 may be used to indicate a sub-analyzer, a cascadeanalyzer, or a tree analyzer element and likewise, the term synthesizerreferred to in FIGS. 2–8 may be used to indicate a sub-synthesizer, acascade synthesizer or a tree synthesizer element, each of which havebeen described above in conjunction with FIGS. 1–1D. Each of theanalyzer and synthesizer elements may be provided according to thematrix transformation technique or alternatively each of the analyzerand synthesizer elements may be provided according to the three pointequations of the forms of Equations 1 and 2 above, or alternativelystill, each of the analyzer and synthesizer elements may be providedaccording to equations of the form:W′=X(n)−X(n−2)V′=X(n−1)

In the case where analyzers and synthesizers are provided as tree typeanalyzers and synthesizers the matrix vectors C_(i) where i=1, 3, 5,etc. . . . operate on the channels in the bottom portions of the treeanalyzer or synthesizer while the matrix vectors C_(i) i=2, 4, 6, etc. .. . operate on channels in the top half of the tree analyzer orsynthesizer. Thus, if the analyzer 28 having channels 29 a–29 h wereprovided by an 8 dimensional matrix having vectors C₁ through C₈ thenchannels 29 h through 29 e would be operated on, respectively, by matrixvectors C₁, C₃, C₅, and C₇ and channels 29 d through 29 a would beoperated on, respectively, by matrix vectors C₂, C₄, C₆, and C₈.

Referring now to FIG. 2 a signal encryptor 32 includes an analyzer 33having signal encryption circuit 34, which may be provided for exampleas a random number generator, coupled thereto. The analyzer 33 isprovided by coupling a plurality, here five, sub-analyzers as shown.Each of the sub-analyzers may be of the type and thus operate in asimilar manner as the sub-analyzer described above in conjunction withFIG. 1. Each of a plurality of analyzer output ports 33 a–33 f arecoupled to a corresponding one of a like plurality of input ports 36a–36 f of a synthesizer 36. Likewise, the synthesizer 36 is provided bycoupling a plurality, here five, sub-synthesizers as shown. Each of thesub-synthesizers may be of the type and thus operate in a similar manneras the sub-synthesizer described above in conjunction with FIG. 1.

It should be noted that here five sub-analyzers and sub-synthesizers arecoupled as shown, however, those of ordinary skill in the art will nowappreciate that any number of sub-analyzers and sub-synthesizers may beused. It should also be noted, and as described above, that althoughcascade analyzers and synthesizers are here shown, tree-type analyzersand synthesizers may also be used. An input signal X may be decomposed,therefore into any number N of signals. In general, and as mentionedabove in conjunction with FIG. 1, the decomposition procedure performedby the analyzer and thus the number of coupled sub-analyzers may bepreferred to cease when the Nyquist frequency of the Nth signal V_(N) isbelow the known lower cutoff frequency of the input signal X.

In general operational overview of the signal encryption circuit 32, theinput signal X, which may be for example a voice signal, is fed to theanalyzer input port 33 a′ and decomposed by the analyzer 33 in themanner described above in conjunction with FIG. 1. Here, the inputsignal X is decomposed into signals V₁, V₂, V₃, V₄, V₅, and W₁, W₂, W₃,W₄ and W₅ as shown. To encrypt, the random number generator 34, whichmay be provided for example as a digital random number generator, feedsan encryption signal to each of the signal paths of the residuals,W′₁–W′₅. The value of each of the residuals, W′₁–W′₅ are thus modifiedto correspond to signals E₁–E₅.

As shown here one method for encrypting the residuals W′₁–W′₅ is toprovide a logical exclusive or (XOR) operation between the sign of theresiduals W′₁–W′₅ and a logical variable having a value corresponding toa logical one (as shown). Alternatively, another method of encryptionmay be provided by adding several individual secret binary bit streamsto one or all of the residuals W′. The residual signals W′ are generallyprovided having a relatively low power level and thus the residualsignals W′₁–W′₅ may appear simply as noise signals (i.e. “buried” innoise) by either or both of the above mentioned encryption techniques.Other methods of encrypting well known to those of ordinary skill in theart may also be used including permutation of channels and substitutionof symbols. It should also be noted that alternatively each of thesignals V₁′–V₅′ may be encrypted or a combination of the signals W_(K)′and V_(K)′ may be encrypted.

It should also be noted that the signal encryptor 32 may be modified toinclude digital cipher feedback, which is common in the encryption art.To provide digital cipher feedback in the present invention, each of the5 XOR outputs from the analyzer should be fed back to the inputs of 5separate secret random number generators (not shown) in the manner ofthe data encryption standard (DES) for encryption.

It should also be noted that any means well known to those of ordinaryskill in the art for providing random digital numbers including but notlimited to the DES, with or without cipher feedback may be used.Furthermore, other methods of deriving and using the random numbers tomodify the value of the residuals W′ including but not limited toadditive noise techniques and the like which do not use the XOR functionmay also be used.

The encrypted signals E₁–E₅ are subsequently coupled to correspondingones of the analyzer output ports 33 a–33 f and then fed to thesynthesizer input ports 36 a–36 e along with an unmodified signal V5 fedto input port 36 f. The synthesizer 36 employs Equations 3 and 4 toreconstruct the signals E₁–E₅ fed thereto to provide a reconstructedoutput signal Y which may be transmitted on a communication channel (notshown) for example.

A receiver (not shown) receives the transmitted signal and analyzes thesignal to recover the signals V5 and E₁–E₅. The receiver subsequentlydecrypts the signals E₁–E₅ to recover the signals W₁–W₅ andre-synthesizes to get the original signal X.

The synthesizer 36 reconstructs the signals E₁–E₅ such that theresultant reconstructed signal X′ provided at the synthesizer outputport should not occupy more than the Nyquist bandwidth. Thus it ispossible to have the sampling frequency of the highest frequencyencrypted stage correspond to twice the upper cutoff frequency of thetransmission channel.

The reconstructed encrypted signal X′ provided at the synthesizer outputport will of course have a different (noise-like) power spectrum and adifferent average total power due to the additive noise introduced. Theinput signal power may be diminished to restrict the total channelpower. Means for restricting the total channel power may include meansfor transmitting the power factor to the receiver by using the amplitudeof pilot tones which may be used to synchronize the decryptor and anADC. These pilot tones may be used to derive the ADC clock by a phaselocked loop for example and may be provided as explicit narrow bandtones or, alternately the pilot tones may be provided as a secret codedsequence added to the residual W′ to hide the true value of theresidual. The receiver would correlate the appropriate residual in orderto establish time synchronization. Specific coded sequences (such as theGOLD codes and JPL ranging codes) having suitable correlation propertiesrequired for acquiring synchronization by this method are known in theart of spread spectrum communications.

Furthermore the encryptor recovers the signal out of the receiversubstantially exactly like the original input signal except for thepresence of noise from the channel and noise-like fluctuations due toimprecise timing synchronization of the ADC converters in thetransmitter and receiver ends.

The signal encryptor may also include a means for line equalization (notshown). Such means are well known to those of ordinary skill in the artof modem design. Variations in line loss with frequency (tilt) and phaseshift may be compensated by adaptive filtering and often includes aprecursor burst of known energy to set the filter parameters and a meansfor periodically modifying the equalizer coefficients based on a measureof the quality of the received signal.

It should also be noted that if in the signal encryptor 32 the value ofthe input signal X is set to zero, then the signal encryptor outputsignal X′ would correspond to random noise generated by the encryptingrandom number generator. Thus in this case the signal encryptor 32 actsas a broadband random number generator. Thus, the signal encryptor maybe used as a generator to provide a signal on the line which may be usedin the line equalization process.

Furthermore the analyzer may be used to measure the channel response ofeach channel. If the synthesizer, thus provides a test signal on eachchannel the analyzer may measure the response on each channel todetermine the loss on each channel. Thus the analyzer and synthesizermay be used to provide a method of line equalization.

As will be described below in conjunction with FIG. 4, this concept maybe further refined to only using a synthesizer at the sending end andonly an analyzer at the receiving end.

The delay in a tree may correspond toDelay=2^(L)−1where L is the number of levels in the tree. Thus the tree process in afive level tree introduces a relatively short delay of only 31 samples.Likewise, for a cascade system, it is believed that the delay in eachsignal path is provided by the above equation where L now corresponds tothe number of stages in a particular path of the cascade. Since humanhearing is sensitive to echoes from a hybrid coupler found in mosttransmission media, this short delay is a desirable feature.

Referring now to FIG. 3, a transmitting and receiving system 40 for thetransmission and reception of secure signals includes a transmittingportion 40 a and a receiving portion 40 b. The transmitting portion 40 aincludes a signal conditioning circuit 41 including an input filter 42having appropriately selected filter characteristics preferably selectedto provide Nyquist filtering to an analog signal fed thereto. The filter42 couples the analog signal to an analog-to digital converter (ADC) 43which converts the received analog signal into a digital signalrepresentative of the analog signal.

The analog signal fed to the ADC 43 input port may be fed through anamplifier or other preconditioning circuits (not shown) and then fed tothe ADC 43. Preferably signal preconditioning circuits such as low noiseamplifiers and buffer amplifiers are relatively wide-band amplifiers andare further characterized as having relatively low levels of phasedispersion over the bandwidth of the amplifier. That is, the amplifiersimpart to the amplified output signal a substantially equal phase shiftto the amplified output signals therefrom at least over the bandwidth ofthe transmitted signal. Further, the sampling rate of the ADC 43 ispreferably greater than twice the Nyquist sampling frequency (i.e.greater than twice the frequency of the highest frequency componentsignal in the input spectrum).

The ADC 43 converts the signals provided from the filter 42 inaccordance with the predetermined sampling rate to provide a stream ofdigital words. At the output of the ADC 43 such stream of digital wordsare fed to a signal encryption circuit 44.

The signal encryption circuit 44 includes an analyzer 46 whichappropriately decomposes the signal fed thereto into a plurality ofsignals, an encryptor circuit 48 which encrypts the decomposed signalsand a synthesizer 50 which combines the signals fed thereto into areconstructed encrypted signal. The signal encryption circuit 44 may beof the type and operate in a manner similar to the signal encryptioncircuit 32 described above in conjunction with FIG. 2.

The encryptor circuit 44 may encrypt the signal by adding a secretnumber to the residue, W′ with the value of the secret number known onlyto the sender and receiver. Such an addition may typically be donemodulo 2. That is, by providing a logical exclusive-or (XOR) functionbit by bit to the residual signal. Such addition may also beaccomplished by simply adding modulo the actual word size of the data.

The encryptor 48 may be provided as a secret number generator which maybe of a type that provides a different secret number based on the numberof times it is requested to do so, or it may be of the type that theoutput depends only on its input. The latter type may often beself-synchronizing. The former type may suffer loss of sync between theencryptor and the decryptor if the receiver, for any reason, loses bitor word synchronization with the transmitter.

The encryptor 48 may provide permutation operations among the severalchannels or alternatively the encryptor 48 may provide substitutionoperations or alternatively still the signal encryptor may provide thelogical or operation described above in conjunction with FIG. 2.

The signal encryption circuit 44 feeds the reconstructed encryptedsignal to an input port of an output signal conditioning circuit 51which includes a digital-to analog-converter (DAC) 52 which may beprovided as a logarithmic ADC for example, which converts the encryptedbit stream fed thereto to an analog signal representative of theencrypted bit stream. The analog signal is subsequently fed to an outputfilter 54 having appropriately selected filter characteristics asdescribed above. The filter 54 couples the signal fed thereto to a firstend of a transmission channel 56, which may be provided for example as atelephone line. A second end of the transmission channel 56 is coupledto the receiving portion 40 b of the transmitting and receiving system40.

In general overview, the receiving portion 40 b receives a secure signalfed thereto and decrypts the signal to provide a clear text signal at anoutput port. The receiving portion 40 b of the transmitting andreceiving system 40 receives the encrypted signal through an inputsignal conditioning circuit 57 at an input port of an input filter 58having appropriately selected filter characteristics. The input filter58 couples the signal to an ADC 60 which converts the analog signal toprovide a stream of digital words in the same manner as described above.The filter 58 may be provided having a low pass filter frequency cutoffcharacteristic corresponding to one-half the sampling frequency of theADC 60. The ADC 60 feeds the stream of digital words to a decryptioncircuit 61 which includes an analyzer 62 which decomposes the signal fedthereto into a plurality of signals and a decryptor circuit 64.

If the encryptor circuit 48 in the transmitting portion 40 a of thesystem used a secret number to encrypt the signal, the decryptor circuit64 subtracts the secret number, to recover the original data.

The decryptor-synthesizer combination 64, 66 performs the inverseoperation of the encryptor-analyzer 48, 46 at the transmitter and feedsthe decrypted and appropriately recomposed signal to a digital to analogconverter (DAC) 68. The DAC 68 provides an analog signal correspondingto the bit stream fed thereto and feeds the analog signal to a receiveroutput filter having appropriately selected filter characteristics asdescribed above.

If the analyzer and synthesizer are provided as the cascade type or thetree type, then as mentioned above, the number of levels may typicallybe determined by the lower limit of the input signal bandwidth. Thus apractical requirement for most applications may be to dispose apre-filter (not shown) having a high-pass or band pass filtercharacteristic prior to the signal conditioning circuit 41.

In any case, whether such a filter is or is not provided, the system ofsender and receiver may not recover a signal having a bandwidth whichexceeds the bandwidth of the communications link 56. In applicationswhere the communication link 56 is provided by electromagnetic or sonicenergy, the bandwidth of the link generally has practical limits, forexample a single television (TV) channel.

Referring now to FIG. 4 a signal encryption system 72 receives an analogsignal at an input port and filters and converts the analog signal to afirst stream of bits via a filter 74 and an ADC 76 in the same manner asdescribed above in conjunction with FIG. 4. Here, the bit stream is fedfrom the ADC 76 to a signal encryptor 78. Here the signal encryptor 78receives the signals directly on separate signal channels. A randomnumber generator 80 is coupled to the signal encryptor 78 and feeds astream of random bits to the signal encryptor 78. The random stream ofbits modifies the first bit stream to thus provide an encrypted signal.The signal encryptor 78 combines the signals to thus provide areconstructed encrypted signal at an output port.

The encrypted signal is subsequently fed from the signal encryptor 78 toan input port of a DAC 82 which converts the encrypted bit stream to ananalog signal. The analog signal is coupled through a filter 84 whichprovides an appropriately filtered signal to a first input port of asignal combining circuit 86.

A timing circuit 87 includes a timing signal generator 88 for providinga timing signal. The timing circuit 87 may provide to a receiver 92either an analog or a digital timing signal and thus an optional ADC 90is here shown coupled between the timing signal generator 88 and thesumming circuit 86. If the DAC 90 is included in the timing circuit 90then the timing signal is fed through optional signal path 91′ to thedigital input port of the DAC 8 d. If the ADC 90 is omitted then thetiming signal is fed through signal path 91 to the summing circuit 86 asshown.

It should be noted and those of ordinary skill in the art will recognizethat implicit in the recovery of information at the receiving end of ananalog communications link using the analyzer and synthesizer buildingblocks operating according to the above equations is the need tosynchronize the ADC in the transmitting portion of the system and theDAC at the receiving portion of the system such that the transmit andreceive portions of the system agree on the exact time of sampling.Large inaccuracies in synchronization will result in gibberish out ofthe receiver. Small statistical jitter in the timing synchronizationwill have the same effect as noise on the link.

Timing and synchronization schemes are known to those of ordinary skillin the art. In one method for example, an oscillator locked to (i.e.derived from) the sampling rate of the sending portion may be sent overthe link. The received oscillator signal may be used to derive (viaphase locked loop techniques for example) the receiver's clock signalsfor sampling. As mentioned in the secure transmitting and receivingsystem 40 (FIG. 3) the timing information may be applied to theencryptor and decryptor respectively.

In those systems employing cascade-type or tree-type analyzers andsynthesizers (FIGS. 1A–1D) two timing signals may be required becausethe inherent delay in the sending portion of the system effectivelydefines “words” rather than just signal bits that should besynchronized. Such word synchronization may be achieved by providing asecond sender oscillator which may be locked to the first oscillator(alternatively, the first oscillator may be locked to second oscillatorthus requiring only a word synchronization) and operating at a ratedepending upon both the number of levels in the cascade and on the typeof system being provided i.e. a signal encryptor, a signal compressor,or a modem. Such oscillator signals may occupy the same bandwidth as thesystem information, since they may be subtracted out by the knowntechniques for removing a signal of known frequency and constantamplitude. In practice these two oscillators should preferably providesignals having a particular frequency and a particular amplitude. Inpractice these two oscillators should preferably be locked to signalshaving frequencies corresponding to those frequencies which define theextreme upper and lower band edges of the frequency bandwidth of thesystem information (and the system information while still maintainingthe requirement that will not exceed the band limits of the link.

One method for removing an interfering signal (namely, the receivedoscillator signal) from the accompanying system information signal is toform a feedback loop that subtracts out an amount of the known (asreceived) frequency until the resulting difference no longer containsany narrow band correlation to the known (received) oscillator. Theoperation of a phase locked loop includes multiplication of an inputsignal with a local oscillator (which, in turn, is locked to thereceived timing signal) and then integrating the result with a low-passfilter.

Another method for timing synchronization between the sending andreceiving portions of the system will be described below in conjunctionwith FIG. 7.

Referring now to FIG. 5 a system for transmitting digital data over ananalog medium 94 includes a modulator-demodulator (modem) 95 here only amodulator portion of the modem 95 being shown. The modem 95 includes adata assembly unit 96 for forming digital data into frames or byteshaving a predetermined length appropriately selected for datatransmission. The data assembly unit feeds the data to the input portsof a synthesizer 98 which forms the data into a bit stream in accordancewith the technique of the sub-synthesizer 20 described above inconjunction with FIG. 1. That is, here digital data to be transmitted isapplied to the residual inputs W′₁ . . . W′_(N) of the synthesizer. Thisdata may be encrypted by a random number generator (not shown) whichprovides an encrypting sequence which may or may not be a secretsequence.

The synthesizer 98 feeds the bit stream into an input port of a DAC 100,which may be provided having a nonlinear response characteristic, whichgenerates an analog signal corresponding to the bit stream fed thereto.In accordance with the Nyquist sampling theorem procedure, the digitizedsamples out of the sender should be converted via the DAC 100 to ananalog signal representation of the digital signal fed thereto. Theanalog signal is subsequently filtered for anti-aliasing with a filter102 having a low pass filter characteristic and preferably having arelatively steep filter skirt and a cutoff frequency corresponding toone-half the sampling rate frequency. The filtered signal is coupled toa first input port of a summing circuit 104. A timing circuit 106 feedsa timing signal into a second input port of the summing circuit. Thesumming circuit thus superimposes the two analog signals fed thereto.Alternatively, a timing signal may be transmitted via the input to thesynthesizer 98.

The superimposed analog signal is transmitted over an analogtransmission link 107 (e.g. a telephone line) to a receiver 108 wherethe timing signal may be used to provide timing data for the receiversuch that the bit stream may be recovered from the analog signal. It isbelieved that a modem constructed in accordance with the presentinvention may operate at or near the maximum data rate theoreticallypossible for a transmission link based on signal to noise ratio of thelink and Shannon's Law.

At the receiver 108, substantially all noise in the frequency rangeabove one-half the receiver sampling rate frequency should be filteredwith a filter having a low-pass filter characteristic and having acutoff frequency above one-half the receiver sampling rate frequency. Ifthe transmission link 107 is provided as a transmission line over whichsignals having a frequency between 400 Hz and 3200 Hz may be transmittedthen the input signal may appropriately be sampled at a sampling ratetypically of about 6400 bps.

A single tone having a frequency typically of about 3200 Hz may beprovided by the timing circuit and added to the transmitted signal andphase-locked at the receiver as one means of synchronizing a receiverADC clock. Since the tone may be provided having a known amplitude andfrequency, it may be subtracted rather than filtered out at the receiverand thus there is no resultant loss of data.

Likewise a signal tone having a frequency typically of about 400 Hz maybe used to provide a word synchronization for formulating the wordcomprising the input to all the cascade levels. Signals in the frequencyrange below 400 Hz may be used for signaling for line turnaround in asemi-full duplex modem, for sending reverse channel data and networkinformation. Bits per cascade level may be increased until the maximumpower per unit of data which may be transmitted and Shannon's limit arereached.

Furthermore error correcting codes such as M of N codes (M redundantbits out of N) and scrambling signals may be applied to the input datawords as is known to those of ordinary skill in the modem art.

Referring now to FIG. 6, a coded modem 110 using direct sequence coding,in which each data word modulates all the bits in a sequence of codebits, more than one signal may share the link simultaneously. A sharedsignal S₂ may be provided for example, as a voice signal, television(TV) signal or facsimile (FAX) signal or alternatively the shared signalS₂ may be provided from additional modems of the same type operatingwith orthogonal code sequences.

The coded modem 110 includes a coder 112 for providing the codingoperation. The coded signal is fed to a synthesizer 114 which provides arelatively broadband signal having a noise-like frequency spectrum to aninput port of a summing circuit 116. A sharing signal may be fed to asecond input port of the summing circuit. The summing circuit couplesthe signals fed thereto to a first end of a transmission line 117.

At the demodulator portion, a signal tap 118 couples a portion of thesignal transmitted over the transmission line 117 to an optional signalprocessor 124. The signal processor 124 feeds a processed signal to areceiver 125.

The codes of the coder 112 are selected to have good auto- andcross-correlation properties, and thus the modem data may be recoveredeven when the modem operates at low power relative to the shared signal.

To a sharing device, the receiver 125 which may be provided as a TVreceiver for example, the modem signal may appear to be a small randombackground noise. However if the shared signal is coupled to the TVreceiver 125 through the signal processor 124 containing the codesequences, C, then most or substantially all of the modem “interference”in the TV receiver 124 may be removed by known techniques for cancelingcorrelated noise.

The modem 110 includes the coder 112 for providing a coding operation C,and a decoder 122 for providing a correlation operation, C^. In acorrelation operation, data is recovered by digitally integrating theproduct of the received sequence with the stored code, C. A signalsharing the link will typically tend to integrate to near zero since theshared transmission is uncorrelated to the chosen code, C. The optionalinterference cancellation operation for the sharing signal is performedby the signal processor, 124.

A method for timing synchronization between the sending portion andreceiving portion of the system 110, is now described, however beforedescribing the method it should be noted that although the method may bemore clearly explained within the context of the coded modem, the methodmay also be applied, with minor variations, to other systems includingthe signal encryption systems described above in conjunction with FIGS.2–4 above and signal compression systems to be described in conjunctionwith FIG. 9 below.

At least one of the signals, W′_(k), is forced to be a sequence havingknown correlation properties. In a cascade synthesizer, since “word”synchronization is required, the chosen signal W′_(k) may preferablycorrespond to the signal having lowest inband sample rate. It should benoted however, that in a direct sequence coded modem all the W′_(k)s areso coded.

There are many examples of suitable code sequences such as JPL, GOLDcodes and Walsh codes. For illustration and not by way of limitation,the Walsh codes (also known as Hadamard codes) are described. The kernelfor a Walsh code is provided as:

-   -   +1 +1    -   +1 −1

Higher order codes are found by substitution of a level into the kernelas shown

-   -   +1 +1 +1 +1    -   +1 −1 +1 −1    -   +1 +1 −1 −1    -   +1 −1 −1 +1        which may be compactly expressed as:    -   +1 +1 +1 +1=code 1    -   +1 −1 +1 −1=code 2    -   +1 +1 −1 −1=code 3    -   +1 −1 −1 +1=code 4

Many other codes, (such as the GOLD codes) are known that have “good”correlation properties. By correlation is meant multiplication andintegration which in a two valued binary case (+1,−1) reduces to justthe vector dot product of the sequences. A dot product between twoidentical codes provides a predetermined output (i.e. (Code 1) DOT (Code1)=4 ) However, a dot product between 2 unlike codes would provide anoutput of zero (i.e. (code 1) DOT (codes 2, 3, 4,)=0) Similarly, thisrelationship also holds true for each of the other 3 codes. Thus, theseare orthogonal codes.

Non-orthogonal codes having a large auto-correlation and a smallcross-correlation may also be suitable, and some of such codes are knownto be particularly good for fast acquisition of synchronization in asliding correlator. An example of a sliding correlator may be made withcode 3. If word sync is unknown (assume for this discussion that bitsync is known) then one of four possibilities may occur in thereceiver's correlator: they are

-   -   +1 +1 −1 −1 assumed timing of the receiver's word clock    -   +1 −1 −1 +1 received pattern possibility 1    -   −1 −1 +1 +1 possibility 2    -   −1 +1 +1 −1 possibility 3    -   +1 +1 −1 −1 possibility 4

Correlation, that is the dot product of the receiver's code with each ofthe 4 possible patterns reveals that the correlator will compute a dotproduct of 0, −4, 0, and +4 for each of the 4 possibilities,respectively. However, only the correct word synchronization (i.e.namely possibility 4) will have large positive (i.e. +4) correlation. Bysliding the receiver's assumed clock, bit by bit relative to theincoming signal, and performing the correlation, the receiver may thusfind word synchronization, hence the name “sliding correlation”. Thus,it should also be noted that the maximum correlation will occur whenboth the word and bit synchronization are correct.

To accomplish the foregoing method of synchronization, the residual, W′,of the lowest frequency stage in the sender's cascade may be arranged tohave code 3 define its value (or at least the sign of W′ would followcode 3). The preceding discussion illustrates some of the many waysreceiver synchronization may be achieved. In some system applicationssuch as encryption, the timing from the receiver may also drive otherbuilding blocks, such as a decryptor. In full duplex operation, a clockin the receiver may also be used for transmission from that end suchthat there is only one master clocking the system.

Assuming that bit and word synchronization have been established in themodem 110, then an incoming data bit may be expressed as correspondingto either +1 or −1. If that data bit multiplies one of the codes, code 2for example, then the resulting 4 bit sequence is either code 2 or code2 with the sign of each bit reversed. If that sequence is applied to oneof the W′ inputs of a cascade or tree synthesizer as a sequence of bits,then the receiver's analyzer will recover that W′ and may correlate withcode 2 to get either a large positive or a large negative number whichwill determine the receiver output as a +1 or a −1 respectively. Herefor ease of explanation, an example using a single bit has beendescribed. In practical systems of course, such operations wouldtypically be performed on digital words having a plurality of bits.

The application of the coded data bits to the W′ inputs of a cascadesynthesizer is somewhat complicated by the fact that each stage in thecascade operates at a different sampling rate. Such an operation may bemore easily accomplished in a tree synthesizer since input data may beassembled into words and applied all at once at the frequency of thelowest synthesizer stage. For a tree synthesizer, the transmitter'spower would also be distributed more evenly across the link bandwidth—apreferred and efficient case. Spreading the transmitter's energy evenlyacross the link bandwidth is a prescription for possibly achievingoperation at the maximum limit of the link.

Several coded modems of the type described in FIG. 6 may operatesimultaneously over the same link, within the constraint of total linkpower. Each modem should use a different orthogonal code. For example, acode 3 modem would not interfere with a code 2 modem. It should be notedthat the number of modems that may share a link using the 4 bit Walshcodes described above is more than four since each modem may have adifferent and unique combination of codes on each of its independentresidual inputs, provided certain groups of modems don't operate atexactly the same time. It should also be noted that these multiplemodems may simply be provided as different orthogonally coded data bits.

A two wire full duplex modem may be provided by using a companion set ofalmost orthogonal codes. The Walsh codes identified as code 1 throughcode 4 above, and their bit-wise complement, are only one half of the 16possible combinations of 4-bits. As shown below the remainingcombinations also form another set of mutually orthogonal vectors, belownumbered as c5 through c8. This second set of 4 vectors is notorthogonal to the first set. It can be described as “nearly orthogonal”because the dot product of any member of set 1 with any member of set 2is always half the length of the vector; and of course the dot productof any member with another of the same set is always 0 except that theproduct with itself is always equal to the length.

MASTER (set 1) SLAVE (set 2) c1 = + + + + C5 = + − − − = c1{circumflexover ( )} c2 = + − + − c6 = + + − + = c2{circumflex over ( )} c3 = + + −− c7 = + − + + = c3{circumflex over ( )} c4 = + − − + c8 = + + + − =c4{circumflex over ( )}

The master group is orthogonal and the slave group is also orthogonal,however the cross group correlation is −2 for the dual and +2 for anyother cross term. One end of a transmission link may transmit signalsusing the master set of codes and a second end of the transmission linkmay transmit using the slave set of codes. It should be noted that thesame benefits may be realized by using the matrix transformationapproach to provide the synthesizer 114 and analyzer 120.

Thus to provide full duplex operation, the modems at each end of a linkcan be assigned to use set 1 or set 2 as Master and Slave. If the Masterused only codes 1 and 2 and the Slave used codes 3 and 4 then all echosignals would be totally cancelled by the orthogonality but the datathroughout for each modem would be half of the rate possible by usingthe arrangement described above based on the two sets of codes.

Furthermore, coded modems of the type described in conjunction with FIG.6 may coexist with other signals on the link since the correlator willprovide little or no output signal. Longer code sequences may improvethis effect at the expense of lower data throughput. Certain codes,other than the Walsh codes, may be better able to exploit thischaracteristic for multiple access applications.

For a modem 126 as shown in FIG. 7, it is possible to use directsequence code division multiplexing to excite the sub-stages of thesynthesizer cascade. As described above, such a procedure would permitclock recovery based on a sliding correlation. It also allows datamultiplying the code sequence (as is done in direct sequence spreadspectrum) to be recovered at a receive end using correlation techniques.

One application for such a coded modem would be to take advantage of theprocessing gain of the correlation receiver to recover a low powersignal from the modem buried in a large “jamming” signal. Practicalexamples of jamming signals include voice (hence data may be sent as‘noise’ under the voice), television (hence high definition digitalinformation may be sent in the same channel as standard video tomaintain compatibility), code division multiplexing and two wire fullduplex FDX. The disclosed technique is an improvement on those methodsbecause the modem utilizes the bandwidth more efficiently thanheretofore.

Furthermore, the modem described herein, being a modulator (thesynthesizer) and a demodulator (the analyzer) may also take the form ofa baseband RF or soundwave transmitter modulator and receiver (or abovebaseband except for the limits of ADC's). Such a receiver may haveapplication in receiving digital High Definition TV (HDTV).

Coded modems of the type described above in conjunction with FIG. 6 andto be described in conjunction with FIG. 7 below may use Walsh codes topre-encode data which is applied to the w′ and v′ inputs. Because thesynthesizer itself uses Walsh codes, if implemented by the rotationmatrix, it should be pointed out that these two coding schemes areindependent. For example, the code length for the data inputs need notbe the same as the number of synthesizer outputs. The number of outputsis equal to the length of the synthesizer rotation operators. Thus byapplying the matrix method to the coded modem, the coded modem becomesin essence a twice coded modem.

Referring now to FIG. 7, a coded tree modem 126 includes a plurality ofcoder circuits 128 a–128 h coupled to corresponding ones of a pluralityof input ports 130 a–130 h of a tree-type synthesizer 130 operating inaccordance with the principles described above in conjunction withFIG. 1. The synthesizer output port is coupled through a link 132 to aninput port of a tree-type analyzer 134 also operating in accordance withthe principles described above in conjunction with FIG. 1. A pluralityof decoder circuits 136 a–136 h are coupled to the analyzer output ports134 a–134 h to decode the coded signals fed thereto.

In principal the coded tree modem 126 operates such that the codercircuit performs a coding operation of multiplying a data word times anorthogonal code C. The decoder performs a correlation operation heredenoted C^. It should be noted that the final V′ input sequence on line130 h may be set to zero if it is assumed to be below a frequencycorresponding to the lower frequency limit of the passband frequency ofthe link 132.

Referring now to FIG. 8, a system for transmitting and receivingcompressed signals 138 includes a transmit portion 138 a having an inputsignal conditioning circuit 139, which here includes an input filter 140and an ADC 142. The filter 140 and ADC 142 are selected to operate inaccordance with the techniques described above to provide an appropriatestream of digital words to the first input port of an analyzer 144. Aquantizer 146 is coupled between the analyzer 144 and a synthesizer 148.In operation, the signal compressor quantizer 146 maps the residualW′(N) into a new number taking fewer bits to describe. Thus it is acompression operation.

An output signal conditioning circuit 149 includes a DAC 150 coupled tothe output port of the synthesizer 148. The DAC 150 receives a digitalstream of words and provides an analog output signal representative ofthe bit stream fed thereto. A filter 152 having appropriately selectedfilter characteristics couples the analog signal from the DAC 150 to afirst end of a transmission line 154.

A second end of the transmission line 154 is coupled to a receiveportion 138 b of the system 138. The receive portion b includes an inputsignal conditioning circuit 156, which appropriately filters andconverts the analog signal fed thereto to provide an appropriate streamof digital words to the first input port of an analyzer 158. An inversequantizer 160 (i.e. requantizer) is coupled between the analyzer 158 anda synthesizer 162.

In operation, the requantizer 160 remaps to the original bit definition.The compression operation, of course, lowers the information content ofthe signal and that lost information may not be recovered. In manyapplications, however, the lost information is redundant or the humanobserver is insensitive to the level of detail contained in theinformation discarded by the quantizer and thus little or no signaldegradation may be detected.

An output signal conditioning circuit 163 receives a reconstructeddigital stream of words from the synthesizer 162 and provides anappropriately filtered analog output signal representative of the bitstream fed thereto to an the output port of the receive portion 138 b ofthe system.

In speech compression processes, bandwidth may generally be reduced bylimiting the number of bits devoted to the residual W′.

However, an alternative which may provide an increased reduction inbandwidth may be accomplished by first replacing a signal V₅ on thecascade channel corresponding to the lowest frequency band with a zero,thus only transmitting the signal W′₅. Next, a signal W′₃ associatedwith a cascade channel corresponding to the frequency band in the 700 Hzto 1400 Hz frequency range may be eliminated or coarsely quantized.Furthermore, Huffman coding or codebook vector quantization methods maybe used on W′₂.

By adjusting the sample rate, the frequency band from 700 Hz to 1400 Hzmay be isolated. Since human voice especially in the English language,generally does not include a formant in this range, the cascade channelcorresponding to the frequency band from 700 Hz to 1400 Hz may beeliminated (i.e. by setting the residual W′₃ equal to zero) with littleloss in intelligibility. Similarly, as shown in the Table below, W′₁ andW′₅ may also be set to zero.

TABLE SAMPLE RATE SIGNAL (smpls/sec) FILTER RANGE (Hz) W′₁ 5600 W′₁ =0 >>2800 to 3200 W′₂ 2800 W′₂ >>1400 to 2800 W′₃ 1400 W′₃ = 0 >>700 to1400 W′₄ 700 W′₄ >>350 to 700 W′₅ 350 W′₅ = 0 >>175 to 350 V₅ 175 V₅ = 0

Thus, in this example W′₁ through W′₅ and V₅ are sampled at the ratesshown in the Table and it is possible to transmit only the signalscorresponding to the residuals W′₂ and W′₄, for example, which havesample rates of 700 and 2800 baud perhaps less than 2 bits for eachresidual W′₂, W′₄ after Huffman coding. Further reduction may also bepossible since W′₂ may simply be considered as another sampled signaland thus may also be subdivided by multi-resolution analysis to furtherreduce the bandwidth.

By way of example, if 1.5 bits are used for the 700 sample/secondresidual and W′₂ is decomposed into 1400 samples/second, 700samples/second, 350 samples/second, 175 samples/second, and 65samples/second at 1.5 bits each then the total number of bits per second(bps) corresponds to 5085 bps to which should be added overhead bits forframe synchronization. This method is considerably less computationallycomplex than methods such as Linear Prediction Coding 10 (LPC10) anddynamic Excitation LPC and refinements thereto as are known to those orordinary skill in the art of voice compression.

Although not here shown, the receiver may be provided having the sameform with received signal applied to V₀ and the clear output taken fromV′₀.

In view of the above, those of ordinary skill in the art will nowrecognize that combinations of the system described above may be createdto form, for example, an encrypted data compression system for use onanalog links. This may be particularly useful, for example, in thoseapplications such as transmission and reception of high definitiontelevision signals in which the amount of digital data to be transmittedexceeds the Shannon Law limit of the link. Thus, in such applicationsthe data may first be compressed by any algorithmic means until thesample rate is consistent with the Nyquist sampling theorem limit of thelink and the data may then be applied to one of the modem systemsdescribed hereinabove in conjunction with FIGS. 5–7 for example.

Referring now to FIG. 9, a digital compression circuit 166 includes ananalyzer 168 coupled to a quantizer 170. A digital signal is fed to theanalyzer input. The analyzer 168 decomposes the signal and the quantizer170 performs a compression operation as described above in conjunctionwith FIG. 8. The quantizer 170 subsequently feeds a compressed digitalsignal to a first end of a digital link 172. A second end of the digitallink 172 is coupled to an input port of a requantizer 174. Therequantizer 174 receives the signal fed thereto performs an inversequantization process and subsequently feeds a requantized signal to asynthesizer 176. For a digital scheme it is only necessary to perform alogical operation, such as an exclusive or operation, between the Wsignal and a random number RN.

Referring now to FIG. 10, a telephone modem 178 includes a synthesizer180, capable of operation over the frequency band from 400 Hz to 3200Hz. The synthesizer 180 feeds a signal to a DAC 182 having a samplingrate typically of about 6400 samples/second which converts the bitstream fed thereto to an analog signal. The analog signal issubsequently fed over an analog transmission link 184 to an ADC 186 alsohaving a sampling rate typically of about 6400 samples/second. The ADC186 converts the analog signal fed thereto to a digital bit stream. Thedigital bit stream is subsequently fed to an analyzer 188.

The synthesizer 180 may be provided in accordance with the matrixtransformation methods described above in conjunction with FIG. 1. Itshould be noted, however, that the synthesizer 180 may alternatively beprovided as a three level tree synthesizer in accordance with Equations3 and 4 also described above in conjunction with FIG. 1.

In the matrix approach, the synthesizer 180 operates on the data framesfed thereto with a rotation matrix. Here, an 8 dimensional rotationmatrix would be applied to the data frames. Similarly, the analyzer 188would perform an inverse rotation operation by applying an 8 dimensionalmatrix corresponding to the inverse matrix of the matrix used by thesynthesizer 180.

As mentioned above, the basic sampling rate of the DAC 182 and ADC 186is 6400 samples/sec. The digital input and output operate at a framerate of ⅛ or 800 frames per second. Each frame may be composed, forexample, of 35 bits divided into 7 words of 5 bits (or more) each andapplied to channels 180 a–180 h. The throughput is 28,000 bps (35 bitstimes 800 frames/second). The usable rate is determined by the S/N ratioon the link 184 and forward error correction (FEC). Thus, the modem 178may operate at a data transfer rate at or below 28 kbps.

Furthermore, the modem 178 may utilize FEC and as is common practice,the frequency range from 300 Hz to 400 Hz may be used for frequencyshift keying (FSK) diagnostic signaling.

Channel 180 h corresponds to signaling below 400 hz and cannot be usedfor data in this example. However, if a signal having a constantamplitude and alternating sign is applied to channel 180 h then a 400 hztone may be filtered from the line signal by a receiver (not shown) toaid in synchronization. In addition, the known pre-selected amplitudemay be recovered from channel 188 h as data and used as a gaincalibration signal in the receiver and, may further be used to defineblock boundaries for FEC block coding schemes.

Accurate timing synchronization and gain calibration are important tothe operation of the modem 178. As described above in conjunction withFIGS. 6 and 7, coded modems may obtain synchronization informationwithout any tones appearing in the transmitted signal. Thus, clandestinecommunications by encrypting coded modems may be transmitted as lowlevel, seemingly uncorrelated, noise in the same narrow bandwidth assimultaneous, non secret, communications.

The data in the channels 180 a–180 g may be scrambled, as is commonpractice in modems, so that the output appears more noise-like whentransmitting the commonly encountered input string of 35 zeros or ones.Without scrambling, an input string of zeros will produce a modulatedbut strongly correlated output without a DC component.

The data received by the analyzer 188 will be a multiple of the actualdata. The multiplication factor will correspond to the dimension of therotation matrix which in turn corresponds to the number of channels inthe synthesizer 180. The receiver should quantize, that is, round thereceived channel output to the nearest multiple then divide by themultiple to reduce the effects of noise on the link 184.

A modem using baseband modulation techniques, in which no carrier signalexists, may also be provided by using the matrix rotation approach or bythe equations provided in conjunction with FIGS. 1–1D above. In eitherof these approaches, the modem must demodulate and process groups of twoor more samples to recover the data. In a conventional baseband system,single samples are processed to recover the data.

Thus, in a baseband system a modem operating according to the matrixmethod described above should modulate and demodulate groups of samples.For example, if the matrix vectors are of length eight, then eightsamples should be processed together as an independent group by thedemodulator. That is, the demodulator multiplies the group of samples bythe inverse matrix used in the modulator. It should be noted that thegrouping of samples is part of the modulation and distinct, for example,from a block data coding method for error correction, and bothtechniques may be simultaneously used in a single modem.

In summary, a baseband modem technique uses an invertible mapping.According to the foregoing disclosure, and the issued parent patent,U.S. Pat. No. 5,367,516, the modulation and demodulation mappings may becharacterized as a filter bank synthesizer and analyzer, or a rotationand counter-rotation matrix, or a baseband mathematical transformationand its inverse transformation. These characterizations may be distinct,or in some instances they may describe the same operations in differenttechnology. The terminology and unification of the threecharacterizations appear in publications cited in conjunction with thefiling of the instant patent application. For example, the analyzer andsynthesizer in FIG. 1 may be a two-channel Quadrature Mirror Filter(QMF) pair as described by Vaidyanathan and Hoang. The matrices may becommuting polyphase filter matrices of Viterli and Gall or similarly ofVaidyanathan and his students. Inverse polyphase matrices describe theQMF of FIG. 1 as well as filter banks with more sub-bands, such as thepolyphase banks that are functionally equivalent to the structure inFIGS. 1C and 1D. Finally, the structure of FIG. 1A, which is based on atwo-channel QMF, is mathematically equivalent to the Discrete WaveletTransform (DWT) as described by Rioul and Vitterli. Scaling functionsand residual functions are terms used to describe Wavelettransformations.

The invertible mapping for the modem is similar to a geometric rotation,although the mapping may or may not have a pure delay due to the filter.A sequence of samples X(0) through X(3) may be considered as thecoordinates of a vector X in a 4-dimensional space. A lineartransformation may be made from the coordinate system of X to anothercoordinate system in the same 4-dimensional space. Thus the vector X maybe transformed to a vector Y by a “rotation” matrix, or by a filterbank, or by a mathematical transformation such as the DWT.

The components of vector X can be equated to a frame of successivesamples into the transmitter's D/A converter. The components of Y areassigned to those inputs to the filter bank corresponding to thesub-bands that are within the transmission bandwidth. In a modem basedon the foregoing disclosure, and the issued parent patent, U.S. Pat. No.5,367,516, X and Y are two different coordinate representations of thesame vector. The modem modulator can be considered as a linear“rotation” operator [M] and the demodulator as an operator [D]. At thetransmitter, it follows that X=[M]Y. The vector Y is being sent so thereceived data [D]X=Y if the modem is correctly conveying the data, acondition met by the requirement that [D][M]=z[I], where [I] is theidentity matrix and z represents a pure, frequency independent delay, ifany, through the system. If there is additive noise at the receiver thenthe linear demodulation operator must remove the noise term by anon-linear threshold operation as is common in all modems.

In a general sense that includes both linear and non-linear operators,all modems require that [D][M]=z[I] in order to recover data correctly.However, in the baseband modem of the foregoing, and in the issuedparent patent, U.S. Pat. No. 5,367,516, the geometric rotation analogyresults because its linear operators commute. That is [D][M]=[M][D], orat least they can very nearly commute over the entire useful bandwidthof the channel. Geometric rotations in 2-dimensions commute androtations in any dimension can be made to commute by making a series of2-dimensional rotations that commute with a reversed sequence ofcounter-rotations, which is the essence of Vaidyanathan et al.'s designprocedure for multi-dimensional (multi-band) QMF banks of the citedpublications.

As described in the foregoing, and in the issued parent patent, U.S.Pat. No. 5,367,516, any sampled analog signal can be framed as a vectorA and can be digitally encrypted without bandwidth expansion and withoutdigital compression since an encryptor at the transmitter can transmitX=[M][e][D]A and the receiving decryptor can compute A=[M][d][D]X, wheredigital encryption [e] and decryption [d] satisfy [d][e]=[I] andassuming [M] and [D] commute.

Passband modems modulate one or more carriers with one or more separatebaseband modem waveforms that represent data. The carrier modulation andsubsequent demodulation is most often a linear operation in contemporarymodems. Thus, at some carrier frequency, a modem may add a sine andcosine carrier wherein each carrier has been linearly amplitudemodulated by baseband data modulators, or filters. The resultant signalmay have both phase and/or amplitude variations as in a QuadratureAmplitude Modulation (QAM) modem. The baseband modulation of theforegoing disclosure can be applied in this same manner to make a linearmodem for passband.

Non-linear modulations into passband are also possible for data modemsand analog signal encryptors. Non-linear modems have seldom been usedexcept in Frequency Shift Keying (FSK) modems, which are veryinefficient. However, non-linear modems employing the methods describedherein may be more bandwidth efficient than any linear modulationincluding QAM when operated, for example, in the region of interest forwireless communications, assuming for comparison that there is noForward Error Correction (FEC) for either the non-linear or lineardesigns. An exemplary non-linear FM modem is described herein.

An FM DSB signal can be generated by the quadrature carrier methodwherein a sine and cosine carrier are amplitude modulated, respectively,by the sine and cosine of the integral of the baseband modem signal.Thus, a cosine function amplitude modulates the cosine carrier, etc., sothat the FM signal is indeed a non-linear modulation and distinct fromlinear passband modulations, which may or may not include basebandintegrators. FM, and nonlinear Phase Modulation (PM), can also begenerated by a voltage controlled oscillator (VCO) and by othertechniques. The viewpoint taken in the foregoing, and in the issuedparent patent, U.S. Pat. No. 5,367,516, that data and modulated signalscan be different coordinate representations of the same vector, providesinsight and specific techniques for the design of non-linear modemsusing rotations by filter banks or wavelets.

In FIG. 11, an FM modem 200 according to the present invention isillustrated in block diagram form. The modem is comprised of a transmitportion 202 and a receive portion 204. Input data into the transmitportion 202 is first divided into a data vector representation by apartitioning element 206. After non-linear pre-emphasis amplification207 to provide equivalent average power across the partitioned signal,the data vector is rotated to a signal vector by a transmit rotationoperator 208, as disclosed in the foregoing. After non-linearcompression 209 to improve modulation gain (discussed subsequently), thesignal vector is FM modulated as illustrated in block 210 and providedto a transmitter (not illustrated) by the transmitter interface 212 asan output signal.

Typically, the output signal from the transmit portion 202 of one modem200 will not be received by the receive portion 204 of the same modem200. However, it is possible that a transmission path over which theoutput signal is transmitted and from which the input signal is receivedincludes a memory device for storing the transmitted signal. In suchcase, it is possible that the same modem will carry out both thetransmit and receive functions.

For example, in alternative embodiments of the present invention, thememory device is a magnetic disk or a non-volatile solid-state memorydevice; modulated digital information to be stored in analog form islater retrieved and demodulated by the same modem. The memory devicebehaves as a modem transmission link with an exceptionally long linkdelay, so the bit-error-rate calculations and maximum system transferrate into or out of the memory can be determined as for any modemsystem. In particular, the maximum error-free bit rate for recovery ofthe data is given by Shannon's law, where the signal-to-noise rationdepends on the device physics as well as the noise figure of theelectronics.

On the receive portion 204 side of the modem 200, an input signal froman FM receiver (not shown) is received at a receiver interface 214, andis forwarded to an FM demodulator 216. The demodulated signal, afternon-linear de-compression attenuation 217, is then counter-rotated by areceive rotation operator 218. After complimentary de-emphasisamplification 219, the result is assembled 220 into output data which isideally the same as the original input data. The de-compressionattenuation 217 further comprises an equalizer in an alternativeembodiment.

Modems employing convolutional rotation, such as discussed in theforegoing and in the parent patent, U.S. Pat. No. 5,367,516, are optimalin the sense of potential bandwidth efficiency. Encryption using thistechnique is optimal in the sense that the signal encryptor cantransform any band limited analog signal into the digital domain,digitally encrypt it, and then transform the signal back to the analogdomain without changing the analog bandwidth. This is in contrast tocontemporary digital voice encryptors which rely on digital voicecompression algorithms to achieve encryption without bandwidthexpansion. Both of these optimal properties are a consequence of theinvertible baseband transformation used in the modem disclosed in theforegoing and in the parent patent, U.S. Pat. No. 5,367,516. A preferredembodiment of this invertible baseband transformation is the QuadratureMirror Filter (QMF) bank, also known as the wavelet filter, which canprovide lossless reconstruction of signals over all of the filter'spassband. Since the transition region to the stopband at the band edgescan be designed to vanish as the filter delay increases, the modulatorsatisfies Shannon's criterion for no entropy loss in the mathematicallimit.

The rotation operator in a first embodiment is a multi-rate waveletfilter-bank. Such filters are designed in a manner similar to sub-bandcoding schemes. Each analyzer input corresponds to one of M overlappingsub-bands. Ports corresponding to out-of-band regions of the signalspectrum are not used for data, though they are used in a furtherembodiment to carry baseband synchronizing bits without violatingspectral constraints. Regardless of the number of sub-bands, thepolyphase rotation matrices commute exactly over the entire band. Thereis a very small entropy loss at the band edge (but not betweensub-bands), as determined by the independently chosen filter length.

For a time-frame corresponding to M samples, the input binaryinformation is partitioned into the integer coordinates of aninformation vector. The output of the demodulation operator may beviewed as a vector with non-integer coordinates. “Thresholding” is anon-linear operation that removes noise by rounding the received vectorcoordinates into integers.

It is normal modem practice when transmitting data to encode the data asequally spaced signed odd-integer values. For example, two bits areencoded as one of four levels, −3, −1, +1, +3. In this case,thresholding rounds to the nearest odd-integer value. The modem can alsobe used to transmit any bandlimited analog signal which has beendigitally processed by sub-rate filtering, including but not limited tothe digital processing step of digital encryption as described in theforegoing and in the parent patent, U.S. Pat. No. 5,367,516. When themodem is being used in this manner, the input samples to the modemmodulator may be considered as signed integers spanning even and oddvalues including zero. Receiver thresholding rounds to those values. Tobe consistent in notation, integers are assumed to be input to themodulator, the modulator sends integers to the D/A converter and thereceiver obtains integers from the A/D. The demodulator outputs integerswhich are thresholded by appropriately rounding the quotient to integersor odd-integers after dividing by the rotation matrix normalizationconstant. As described in the foregoing and in the parent patent, U.S.Pat. No. 5,367,516, this rotation gain is computed from the rotationoperators (the filter bank polyphase matrix).

The transmitted data, including synchronizing information, isrepresented by an M-dimensional vector. In one embodiment, the moduloaddition of a secret vector to the carrying vector in the datarepresentation results in a highly secure, digitally encrypted, analogsignal out of the D/A. Decryption modulo-subtracts the secret vector torecover the carrying vector.

Input information bits can be assigned to the data-representationcoordinates of a vector and multiplied at the chip-rate by apseudo-random sequence unique to each coordinate. The chip-rate resultis then input to the modem's inband data ports where it undergoesup-sampling by a factor equal to the dimensionality of the rotation.Modem transformation results in an orthogonal summation of eachup-sample-filtered subsequence into overlapping sub-bands of thetransmitter's spectrum. Alternatively, a single spreading function isapplied to the information, which is then partitioned according to thedesired modulation efficiency into the components of a vector, which isthen transformed by a wavelet filter bank.

A cooperating receiver filters the input into sub-bands bycounter-rotating the signal vector back into data space coordinates.However, the effect of the spreading function must be removed prior tothresholding since the spread signal may have a signal-to-noise ratio ofless than 1. “De-spreading” is accomplished by correlating the knownspreading functions with the unthresholded data-representationcoordinates. Since any vector in the transmitter's data-representationcan be re-oriented by an invertible secret transformation (i.e.digitally encrypted by modulo addition of a secret vector) to anywherein the in-band signal space, a truly secure spread-spectrum can be made.The receiver must decrypt the unthresholded data-representation prior tode-spreading. The receiver counter-rotates, decrypts, de-spreads, andthresholds, in that order. Geometrically, the encrypted vector is alwaysinside a radius R, the radius of the N-dimensional signal space.Modulo-R vector subtraction preserves the large non-integer noise vectoradded onto the encrypted carrying vector so it may be subsequentlycorrelated (de-spread) and thresholded.

Thus, to reiterate, an N-dimensional space is defined by N samples intoa D/A converter. The data coordinate system is chosen so that a subset,n of the N coordinates in the data representation, define all theinformation points that are in-band. For each of the n coordinates, Bbits of data can be sent as one of 2^(B) integer levels with anefficiency of 2*B bits/Hz per coordinate (i.e. per sub-band). Since thesub-bands overlap perfectly inside the passband and have typically −70db stop-bands outside, the overall bandwidth efficiency is nominally 2*Bbits/Hz if the filter transition region is negligible compared to thetotal bandwidth used for data. The remaining N-n data coordinates arenot used for data, but can be used for synchronizing the modem.

A data vector with integer coordinates is therefore rotated to signalcoordinates by a first commuting operation. From a geometric standpoint,this operation can be the application of a polyphase matrix for mapping(or rotation) of a data vector into a signal vector representation.Otherwise, the mapping can be implemented via the use of a FiniteImpulse Response (FIR) filter bank or an Infinite Impulse Response (IIP)synthesizer. Alternatively, the mapping can be a wavelet transformation,which can be shown in several cases to be mathematically equivalent tothe filtering performed by a synthesizer. The first commuting operatorin the transmitter is complimented by a second commuting operator in thereceiver. Together, these operators result in an identity matrix,enabling the complete recovery of input data. Practical FIR latticefilter implementations of a Perfect Reconstruction (PR) QMF bank canresult in an identity matrix multiplied by a scalar gain and a puredelay. The gain factor arises because the matrices are typically inun-normalized integer format, and the pure, frequency independent delayfactor represents the delay through the FIR filters. A Near PerfectReconstruction (NPR) filter bank is also known to those skilled in theart of QMF design. The FIR lattice QMF is designed by employing thecommuting properties of geometric rotations. The FIR transversal filterform is then obtained from the lattice form but in doing so, the PRproperty is compromised and the filter bank provides NPR. However, byusing computer aided design techniques, the NPR filter can usually beoptimized to provide an NPR filter having better overall properties forstopband attenuation, for example, than the PR lattice filter used toinitiate the iterative design. Therefore, the mappings used to constructthe baseband modem modulator and demodulator of the foregoing disclosureand the parent patent application, U.S. Pat. No. 5,367,516, commute inthe sense that it may be a perfect or a near-perfect reconstruction witha possible delay and gain.

A two-dimensional QMF as in FIG. 1 is described in the literature bypolyphase matrices. For example, if the high and low-pass filters areFIR, then each filter transfer function is factorable into its even andodd powers of the z-transform variable. So, if H(z) describes thehigh-pass filter and L(z) describes the low-pass filter, then thefilters of a QMF analyzer are factorable into:H(z)=He(z ²)+z ⁻¹ Ho(z ²)L(z)=Le(z ²)+z ⁻¹ Lo(z ²)

This can be written in matrix form as:

$\begin{bmatrix}H \\L\end{bmatrix} = {\begin{bmatrix}\left. {< {He}} \middle| {< {Ho}} \right| \\\left. {< {Le}} \middle| \;{< {Lo}} \right|\end{bmatrix}\begin{bmatrix}1 \\z^{- 1}\end{bmatrix}}$

The sample rate change in a QMF combines with the delay column matrix onthe right to become a serial/parallel conversion for the analyzer, andthe 2×2 matrix on the right, whose elements are shown as row vectors, isthe polyphase rotation matrix. The sample rate change also converts thepowers of z² to powers of z, but since only the coefficients of powersof z in the row vectors are relevant to calculations, the z variable isoften omitted. A similar definition is used for the QMF synthesizer. Theup-sampling at the input to the 2-band synthesizer implies that only theodd coefficients contribute to the odd output sample. Thus, thefiltering can proceed at half the rate using filters of half the length,which is what the polyphase notation describes. Polyphase matrices forfilters with more than two sub-bands are defined in a similar manner inthe cited publications associated with the parent patent, U.S. Pat. No.5,367,516.

The polyphase matrix may be further factored into a form correspondingto an FIR transversal filter that operates on vectors instead ofscalars. That is, square matrices C_(j) having scalar elements eachmultiply the power j power of z. Thus, the modulation operator [M] isdescribed by:M(Z)=C ₀ +C ₁ Z ⁻¹ +C ₂ Z ⁻² +. . . +C _(L) Z ^(−L)In this form, the modem frames the data bits and converts them to thecoordinates of a vector. The vector is input to the vector-filter toperform the rotation. There are L−1 previous data vectors stored in thedelay line of the transversal filter. The vector filter maps the currentand L−1 previous data vectors to new vectors using the matrices C_(j)and then finds, by vector addition, the resultant baseband modem outputvector. The receiver counter-rotates to recover the data using thevector filter corresponding to [D]. In one embodiment, a hardware ASICuses a single time-shared nine-tap filter with a ROM bank ofcoefficients and a bank of shift registers to implement the modemrotations.

The vector-filter arrangement is of course mathematically equivalent tothe other forms of a rotation. However, the transversal form suggeststhat a modem as described in the foregoing discussion and in the parentpatent, U.S. Pat. No. 5,367,516, can be equalized for distortions in thetransmission link by using the same methods as are applied totransversal equalizers in the known modem art. Significantly, theequalizer filter and the demodulation filter [D] can be one and thesame.

Therefore, a further embodiment of the present invention employsadaptive adjustment of the receiver polyphase matrix, also referred toas a de-rotation matrix. If it is determined that some form of frequencydependent distortion is being introduced into the transmission path, thepolyphase filter in the receive portion of the present FM modem, or anymodem using the methods of as previously discussed and found in theparent patent, U.S. Pat. No. 5,367,516, can be adjusted to compensatefor such distortion. Another way of stating this is that some of thede-rotation coefficients in the de-rotation matrix are adjusted tocompensate for the frequency-dependent distortion in the transmissionpath. The adjustment would minimize a computed error function. This issimilar to the design procedure for optimizing NPR in the design of thefilter bank. In the modem, however, the NPR solution does not accuratelyreconstruct the information because of amplitude and delay distortionsin the modem transmission link. The receiver polyphase filter bank in anFIR implementation is equivalent to FIR filters for each sub-bandwherein the filter outputs are decimated and thresholded to recover thedata. This is the same form as a fractionally spaced equalizer, which isknown to those of ordinary skill in the art of modem equalization.Therefore, the methods in the modem art, such as, but not limited to,the Least Mean Squares (LMS) algorithm, for adaptively adjusting thecoefficients of an equalizer in proportion to an error function can beapplied to the polyphase matrix coefficients of the demodulator toprovide the receiver with an adaptive de-rotation that corrects for linkdistortions without a separate equalizer filter. Thus, a fractionallysampled FIR filter is provided. The adjustment of the receiver polyphasefilter can be done at the initiation of a communication and left at acertain setting, or can be adjusted dynamically by monitoring the RMSspread of noise about the integer or odd-integer threshold values out ofthe demodulator, as one example.

Additive noise in a transmission link between a transmitter and areceiver will add a noise vector to the transmitted signal so that acounter-rotation performed in the receiver will yield a recovered datavector with non-integer coordinates. In a first simplified embodiment ofthe present invention, a threshold operator takes the nearest integercoordinate values as the most likely symbol. In a more sophisticatedreceiver embodiment, thresholding is executed according to a Viterbialgorithm since the rotations are convolutional with M-n extra degreesof freedom. This enables “free” error correction without the use ofparity bits.

In each transmitted frame of a modem according to the foregoingdisclosure and the parent patent, U.S. Pat. No. 5,367,516, each D/Asample depends on the scaling input and all other residual inputs to themodulator. To avoid aliasing in the conversion to and from analog by D/Aand A/D, the highest frequency sub-band is not used to transmit data ina preferred modem design. Since the number of samples to/from theconverter equals the total number of sub-bands M, there is redundancy inthe transmitted signal. Other sub-bands may optionally be omitted toshape the output spectrum, for example to avoid transmitting DC, furtherincreasing the redundancy to n out M samples. Sync signals transmittedby the technique of alternating the sign of a fixed amplitude into, forexample, the highest frequency sub-bands carry no information and do notincrease the redundancy. If the receiver has thresholded a sequence offrames prior to the current frame without bit errors, it may use thoseresults to aid in demodulating the current frame. For example, theunthresholded demodulator output may be midway between two allowedodd-integer threshold values. To decide which soft decision value ismost probable, the receiver could use the prior data frames along withthe two possible current-frame soft decisions to generate in thereceiver a modulated signal to correlate with the actual receivedsignal. Then, the receiver may make a final decision about which integerlevels were most probably transmitted for the current frame based uponthe better correlation. This procedure, which may be implemented throughdynamic programming via the Viterbi Algorithm, is possible because ofthe redundancy inherent in the convolutional output of the modulator ofthe foregoing disclosure and the parent patent, U.S. Pat. No. 5,367,516.It differs substantially from the soft forward error correction (FEC)method known in the modem art as Trellis Coded Modulation (TCM).

In TCM, as commonly practiced, redundancy is provided by appending oneor more suitably calculated parity bits to the data before transmission.The coding gain from the Trellis Coding must make up for the loss inenergy per bit due to the transmission of the extra parity. Furthermore,a TCM modem cannot be optimal because of the Shannon capacity sacrificedto send the parity bit. This does not rule out the use of TCM with thecurrent invention. Other methods of forward error correction such as themulti-dimensional coding methods known in the modem art are alsoapplicable to the inherently multi-dimensional signal generated by acoordinate rotation. Thus, a data modem based on rotations that wouldnormally use only odd-integer coordinates for the data vector couldinstead select the coordinates of successive vectors from a larger setof even and odd integers in a way that decreases the probability of abit error.

Implicit in both the modem and encryptor definitions given above is theassumption that the data will rotate to an N-band analog signal. The Nsamples of the D/A are within a precise time frame. Placing a strictsimultaneous requirement on the bandwidth must be done with careaccording to known transformation techniques since sine and cosine-basedtransformations are complicated by the infinite extent of thosewaveforms. When the time is constrained exactly, the frequencycomponents extend to infinity and vice versa.

Wavelet theory provides for transformations based on “mother wavelets”which do not have infinite extent in time. Like Fourier analysis, manybasis functions are summed together to represent any arbitrary signal.The mother wavelets are stretched and shifted in time to form a set ofresidual basis functions and a related set of scaling basis functions.Thus, a single mother wavelet generates a transformation just as aprototype sinewave generates the Fourier transformations. There areinnumerable “mother” wavelets, each generating a differenttransformation. The wavelet transformations are entirely real (there areno imaginary components or carriers). This halves the complexity ofmodulation and equalization. Since only real components are involved, itis thus possible to frequency modulate an input data stream withouthaving to account for an imaginary component.

An exemplary FM modem according to the present disclosure employs an8-dimensional polyphase filter as the baseband input to an FM modulator.As illustrated in the following table, a non-linear distribution of bitsper band is employed.

PRE-EMPH DE-EMPH BAND BITS LEVELS SCALE FCT GAIN FCT i = B = (L) Hi =(db) Gd = 0 4 +/−1, 3, 5, 7, 9, 11, 13, 15 x1 18.6 1 3 +/−2, 6, 10, 14x2 16.2 2 2 +/−4, 12 x4 17.9 3 1 +/−9 x9 22.0 4 1 +/−9 x9 19.9 5 1 +/−9x9 18.1 6 1 +/−9 x9 16.7 7 0 not used for data (sync)

Thus, in the presently illustrated embodiment, there are a total of 13bits per input symbol. One design goal is that each band is to haveapproximately the same power (here, RMS average=9 db). From thefollowing equations, it is apparent that the noise coming out of the FMreceiver depends on the square of the sub-band frequency. The powerspectral density, or PSD, is given by Couch, “Digital and AnalogCommunications Systems”, 4th ed., MacMillan, equation 7–125:PSD=[(K/A)² ][N ₀ ][f ²]where K is the FM detector gain, A is the carrier amplitude and N₀ isdouble-sided noise power spectral density.

According to this frequency versus noise relationship, fewer and fewerbits may be transmitted within each sub-band as the sub-bands move awayfrom DC since the noise increases as the square of the frequency.Therefore, it is possible to send a larger number of bits at the lowestfrequencies since there is significantly less noise coming out of thediscriminator within the receiver then at the higher sub-bands. In theabove example, sub-band 0 has 4 bits, while sub-band 6 has 1 bit; thenumber of levels in each band are chosen to match the parabolic noisedensity function, PSD, of an FM discriminator.

It is possible in a first embodiment of the present invention to provideto an FM modulator sub-bands having decreasing average power levels withincreasing sub-band frequency. For example, the higher sub-bands cancarry fewer bits but all sub-bands use the same level spacing per bit.In a second embodiment of the present invention, the levels representingeach bit or bits in the sub-bands are spaced apart or pre-emphasized toprovide equivalent average power across all sub-bands. In other words,voltage levels of +/−1, 3, 5, 7, 9, 11, 13 and 15 are employed insub-band 0 in order to represent the data conveyed by the 4 bitsassigned to this sub-band. Voltage levels of +/−9 are used to representthe 0 or 1 state of the lone bit assigned to sub-band 6. The averagepower for these two sub-bands is thus approximately the same. Thistechnique of spacing the levels apart at higher frequency sub-bands isreferred to as a pre-emphasis technique. With a signal pre-emphasized inthis manner in its data coordinate representation, it can then beconverted to its signal coordinate representation by a polyphasecoordinate rotation filter. This sampled analog signal can then be usedas an input to an analog FM transmitter. A receiver detects these levelsand converts them back to bits. Thus, no explicit pre-emphasis orde-emphasis filters are needed.

In the above example, the levels for each sub-band are given as integers(i.e. +/−1, 3, 5, . . . ). In other words, each sub-band has 2^(B)levels. In a further embodiment of the present application, the numberof levels in a given sub-band is not a power of two, but if a total of Kbits are transmitted in p sub-bands, then the total number of levels inp sub-bands is 2^(K). A binary mapping algorithm is then employed fordetermining which of the two to the n levels is represented. Adjustingthe bits per sub-band and total number of bits per symbol enables theoptimization of bandwidth for carrier to noise ratio. Computer aideddesign iterations provide these optimized values.

In order to compensate for the pre-emphasis of the higher frequencysub-bands, it is necessary to “de-emphasize” the received signal at thehigher frequencies in the receiver. Less data is sent in higherfrequency sub-bands, though the data is conveyed at higher levels toprovide uniform power out of the transmitter across the sub-bands.Attenuating the high frequency sub-bands results in de-emphasis gain Gd.This de-emphasis results in an overall gain which, for M dimensions, maybe approximated by the following formula:Gd=Gs×M ³/(M−1)where Gs may be one in a carefully designed modem, or may vary slightlyfor each sub-band in a design having the levels in each sub-bandrestricted to 2^(B) levels. Clearly, a larger number of sub-bands Mincreases the gain. Further improvements in gain are possible by forwarderror correction and by reducing the peak to average power ratio PARusing either companding or controlled vector filtering. The latter is atechnique whereby the output peak voltage is pre-calculated at thetransmitter and extra bits, carrying no useful information, are sentalong with the data. These extra bits are chosen in a manner thatreduces the peak-to-average voltage ratio (PAR) out of the basebandmodulator.

Designating H(i) as the pre-emphasis amplification in sub-band i:Gs(i)=H(i)²/(3i ²+3i+1)

-   -   for sub-bands i=0 to M−1        where the denominator in the preceding expression for Gs is        proportional to the integral from sub-band i to sub-band i+1 of        the PSD of the noise out of the discriminator. Sub-band i=0        carries k bits of data and no pre-emphasis so H(0)=1. Sub-band 1        carries k−1 bits with H(1)=2, etc. A typical assignment for M=8        results in nearly equal power in each sub-band:

i 0 1 2 3 4 5 6 7 bits 6 5 4 3 3 3  2 0 H(i) 1 2 4 8 8 8 16 1for a total of B=26 bits per symbol. The various bit level assignmentsare as illustrated above in the previous example of bits per sub-band.

The total gain G is derived from the de-emphasis gain Gd, as givenabove, the normal FM transmitter gain Gm (also known as the modulationgain), which is due to the FM index, as given below, and thenoise-reduction gain Gr. The total gain G is given by:G=Gm×Gd′×Grwhere G′ equals the minimum de-emphasis gain Gd from the de-emphasisgain formula given above. The textbook modulation gain factor Gm is alsogiven by:Gm=6(m+1)(m ²)/(PAR)²where PAR is the peak-to-RMS voltage ratio into the frequency modulatorand m is the modulation index.

The FM index is the ratio of the peak carrier frequency deviation to thepeak frequency of the baseband signal. As shown in the foregoingformulas, as the peak-to-average ratio (PAR) increases, the FM gain, ormodulation gain, decreases. To address this problem, a furtherembodiment of the present invention employs non-linear amplification ofthe baseband signal prior to insertion into the FM transmitter andcomplimentary de-amplification at the receiver, collectively referred toas companding. In a preferred embodiment, a logarithmic Mu-law functionperforms the transmitter amplification digitally by table look-up. TheMu-law is applied to the signal coordinate representation. In anembodiment preferred for large carrier to noise ratios (CNR), the databits are framed and expressed in data coordinates, then pre-emphasized,then rotated, then Mu-law amplified by look-up before the application tothe FM modulator as a sampled sequence. As previously noted, in oneembodiment, the de-compression attenuation includes an equalizer.

In another embodiment preferred for small CNR, the data is framed andexpressed in data coordinates, rotated to signal coordinates, digitallyMu-law amplified, rotated back to data coordinates, pre-emphasized,rotated to signal coordinates and then applied to the FM modulator. Thislatter configuration, which permits de-emphasis immediately after thediscriminator, may be preferred when operating near FM threshold or whenoperating in the presence of high-amplitude non-Guassian interference.The Mu-law function preferred for the transmitter non-linearamplification is:Vo=SIGN(Vi)×Vp×Log 2(1+Mu×ABS(Vi/Vc))/Log 2(1+Mu)where the input Vi has a maximum voltage value Vc and the output Vo hasa maximum value Vp. The value of Mu is greater or equal to 1 and maytypically be 255. The FM deviation is determined by Vp, whereas Vcreflects the precision of the rotation filter calculations.

A second benefit of logarithmic amplification is a result of thenecessary logarithmic de-amplification (i.e. de-compression) at thereceiver, which amounts to attenuation. The input signal has beenlogarithmically amplified and then de-amplified at either ends of atransmission link. However, noise has been introduced within thistransmission link. Therefore, this noise is inverse logarithmicallyattenuated, resulting in the noise reduction gain Gr, as shown in thereceiver gain equation above.

The bit error rate of a modem depends upon the energy per bit, or inother words, the energy separating each data representation level. Thisrate also depends upon the noise energy density. The signal to noiseratio of an FM modem is:S/N=Gm×Gd′×Gr×C/NThus, unlike linear modems, the efficiency depends continuously on thedesign value for C/N. The efficiency of a linear modem varies in stepsas one adds modulation levels, for example in going from 4-PSK to 8-PSK.For a given design, any increase in C/N improves the bit error rate(BER) for both linear and non-linear modems. However, a small increasein C/N for the FM modem can be used instead to increase the bandwidthefficiency or to reduce the bandwidth at the same efficiency. That is,the increased C/N can be used to reduce the FM modulation index, whichlowers the bandwidth by Carson's rule, which specifies the bandwidth Wof FM or non-linear PM, given by:W=2(M+1)wwhere w is the baseband bandwidth. Thus, a non-linear modem system couldreduce the adjacent channel interference (ACI) by using the excess ofC/N over the design C/N value while maintaining a constant BER andefficiency, or the modem can increase the efficiency at the same BER andACI.

The signal to noise ratio of a modem is Eb/N0 times the bandwidthefficiency of the modem (given in bits per second per hertz), or:S/N=(Eb/N0)×(eff)where Eb/N0 reflects the energy per bit versus noise energy density.

The foregoing application of FM transmission techniques to datatransmission derives benefit from the fact that amplitude distortion isirrelevant for FM since information is retrieved by the receiver fromthe frequency by, for example, tracking zero-crossings. In FIG. 12, thetransmission of data via FM double side-band is illustrated in flowdiagram form. In step 230, the data to be transmitted is polyphasefiltered as previously discussed. Also as previously discussed, FIRfiltration involves the summation of a given number of previouspolyphase filtrations. This will be discussed with respect to FIG. 14.

It is commonly recognized that frequency modulation can be achievedeither by: 1) directly adjusting the frequency of a carrier; or 2)adjusting the phase of a carrier with a signal which is the integral ofthe information to be transmitted. The use of phase adjustment toachieve frequency modulation is preferred in one embodiment of thepresent invention since chips are available which enable phasemodulation in an entirely digital fashion, thus maximizing thereproducibility of the input signal. Phase adjustment FM is illustratedin FIG. 12 for FM double side-band and FIG. 13 for FM single-side band.

In FIG. 12, the polyphase filtered input is integrated at step 232. Alookup table is employed to digitally simulate a sinusoidal carriersignal in step 236. This carrier signal is then phase modulated usingthe result of the polyphase filtration integration (step 238). The zerocrossings of this signal contain the coded information; by counting zerocrossings (digitally—step 240), the desired information is provided toan RF switching amplifier (steps 242 and 244). In a first embodiment ofthe present invention, the switching amplifier is a Class C amplifier.In a second embodiment, a VCO is employed for generating FM at passband.In a third embodiment, FM is generated using the in-phase and quadraturetechnique.

FM double side-band provides significant advantages over prior artmodems. It provides immunity to noise associated with amplitudemodulated signals. Distortion due to non-linear amplification isavoided. The RF amplifier employed need not be as precise (andexpensive) since amplitude modulation need not be accurately reproduced.It also provides a certain degree of circuit simplicity, since carrierrecovery is not necessary at the receiver.

However, FM double side-band provides half the band-width efficiency ascompared to FM single side-band. Therefore, another embodiment of thepresent invention employs FM single side-band. The flow diagram of FIG.13 illustrates FM single side-band for the present invention. In step250, the input data is polyphase filtered using, for instance, an FIRfilter or wavelet transformation as previously discussed. This step isfurther developed in FIG. 14, discussed below. The polyphase filteroutput is then integrated, and this result is stored (step 252).

A Hilbert Transform processes the input signal, effectively phaseshifting the input by 90 degrees (step 254). The output of the Hilbertfiltration is likewise integrated (step 256). The Hilbert filterintegration is then entered into an exponential lookup table (step 258).

A sinusoidal carrier signal is digitally simulated as in step 260 viathe use of a lookup table. This carrier is then phase modulated usingthe integrated polyphase filter output (step 262).

As with the FM double side-band scenario, the zero crossings of thephase modulated carrier are computed (step 264), and a digital pulsetrain corresponding to the zero crossings is produced (step 266).Finally, in the single side-band case, a switching amplifier whose inputcurrent is proportional to the output from the exponential look-up tableoutput amplifies the digital zero crossing pulse train (step 268).

In either the FM double or single side-band cases, the first steps inthe transmission procedure, polyphase filtering the input data, isachieved as follows. With reference to FIG. 14, the input data is firstpartitioned into a block having B bits, wherein the B bits are furtherseparated into M coordinates of a vector (step 270). In the illustratedflow diagram, B=13 and M=8, though other values for these variables areemployable. Since the de-emphasis gain factor varies according to(M³/(M−1)), an increase in M permits a decrease in carrier to noiseratio. The optimal numbers are determined empirically, and necessarilyimplicate the determination of pre-emphasis across the sub-bands, aspreviously discussed in relation to the de-emphasis gain factor.

A stack is employed in step 272 is order to retain this and L previousvectors. As indicated in the figure, a first embodiment uses L=9vectors. These L vectors are multiplied by an M×M dimensional scalarmatrix to effectuate the initial polyphase filtering (step 274). Asillustrated, the scalar matrix is 8×8. Subsequently, in step 276, thematrix multiplication occurs on the (L−1) previously stored vectors inthe stack using different 8×8 matrices for each iteration, and theresult is accumulated and output (step 278). As noted in step 280, theprevious steps shown in FIG. 14 are then repeated at the symbol rate(R/B) in order to transmit R bits/second.

In all of the foregoing illustrations, it is understood that alternativeprocedures for FM modulating input data according to the presentpolyphase technique are feasible. However, the illustrated proceduremaximizes the proportion of data manipulations which are digital.

With reference to FIG. 15, a procedure for receiving FM transmitted datais illustrated. In step 284, the received signal is low-noise amplifiedand passed through an image filter to eliminate images introduced bymixing the received signal with a local oscillator. Next, in step 286,the filtered receive signal is down-converted to an intermediatefrequency and inband filtered. The zero crossings of the IF are countedin step 288; it is unnecessary to recover a carrier signal or todetermine the phase of a carrier using FM modulation.

The DC filtration of step 290 is useful in applications of the presentinvention which are subject to substantial doppler shift. For modemsused for low earth orbiting satellites, where the doppler shift followsknown orbital parameters, filtering is used in one example in order toeliminate the effects of such DC or low frequency distortion. Anotherapproach to addressing such low frequency distortion is to avoid use ofthe lowest frequency sub-band. The decimation filtering of the same stepeffects a smoothing of the zero crossing count, the latter necessarilybeing at a high rate for adequate signal resolution, but which mayintroduce unwanted noise. In step 292, polyphase filtration involves theapplication of the commuting operator employed in the first step of thetransmission procedure.

Thresholding is used to differentiate between received levels. Thetransmitted signal was sent at one of multiple, easily distinguishedlevels. However, noise introduced in the transmission link shifts thereceived signals to points intermediate the expected levels. It istherefore necessary in step 294 to assign the decimated receive signalto one of multiple signal levels.

Though carrier recovery is not required using FM data transmission,baseband synchronization requires some form of bit sync recovery (step294). As previously noted, such sync information can be transmitted insub-bands not employed for data transmission. If doppler shift isexpected to be a problem, such as in low earth orbit satellitecommunications, the lowest frequency sub-band may not be used for data,and may be available for sync transmission.

Similarly, the highest frequency sub-band cannot typically be used. Afilter perfect enough to avoid aliasing in the equivalent A to D and Dto A conversion operations does not exist. By transmitting a known bitpattern at the 3 db point between the second to highest sub-band and thehighest sub-band, the sync information is within the bandwidth of thetransmitted signal, but is not taking up usable data bandwidth. Suchsync information can be inserted by sampling a DC signal at a sync bitrate. High and low pass filtering result in a sinewave output at thesample rate, each square wave being at the high or low end of therespective sub-band. Thus, this sync signal, akin to a pilot tone, isorthogonal to all of the data signals.

The specific steps involved in the application of the operator of step292 is illustrated in FIG. 16. Specifically, the sampled input ispartitioned into a vector having the same number M of dimensions as thatemployed in the transmission procedure (step 300). In the illustratedexample, M=8. The result is pushed onto a stack of L vectors (step 302),wherein the present example employs L=9. The first vector on the stackis multiplied in step 304 by the same 8×8 scalar matrix as employed inthe transmission sequence. This is repeated for L−1 previously vectorson the stack, the results being accumulated (step 306). This accumulatedresult provides the coordinates of a resultant vector sum (step 308)which, after the thresholding and recovery of step 294, yields theoriginally transmitted data. The steps of FIG. 16 are repeated at thesymbol rate R/B in order to receive R bits per second (step 310).

The aforementioned steps illustrate that this is, for all practicalpurposes, an all digital FM modem. The only non-digital parts are apassive tank circuit employed in steps 244 and 268, the receiver frontend in steps 284 and 286, and a D to A converter necessary to modulatethe input current to the Class C switching amplifier using the output ofthe exponential lookup table. The transmitter RF power amplifier is infact a digital switch. The filters could also be implemented so that allmultiplications become binary shifts and adds, thus reducing the costand power dissipation of an ASIC.

The previous description and equations generally assume that only one ofM subbands is unused to prevent aliasing in analog converters. Inalternative embodiments, a speed advantage may flow from allowing morethan one sub-band to be unused.

As previously recognized, the instant technique and general structurecan be adapted in a further embodiment for the transmission of polyphasefiltered data using AM modulation. Adaptive rotation of the receiverotation matrix, as previously described, can be applied to an AMmodulated modem which employs the LMS algorithm. For the LMS method, therows of the vector filter tap matrices are corrected iteratively bysubtracting a vector proportional to the error times the input vectorstored in the filter delay line for that tap. The error can be eitherthe difference between the unquantized and quantized output of the modemreceiver (i.e. the error margin), or the error can be the differencebetween the unquantized output and a known training data sequence. Thesegeneral techniques are like those used in modem equalizers except theyare applied to the rotation matrix without a separate equalizer.

As noted above with respect to one embodiment of the presently disclosedmodem, parallel streams of input data are vector rotated usingmulti-rate filter banks prior to transmission over a data link. Underideal conditions, the data link introduces no distortions, the impulseresponses of the filters are matched wavelets and the receiver filterbank behaves as a counter-rotation in a vector space. Ideally, thesampling of each band of the receiver filter bank at the symbol rateeffectively computes the auto-correlation of the incident signal. Toachieve a wavelet transformation whereby the original data is completelyrecovered, a cross-correlation of the received signals with all earlierand later over-lapping wavelets stored within the filter from past andfuture symbols in the same band, as well as in all other bands, must bezero or at least cause negligible “self-interference.” In other words,the cross terms must be orthogonal. Yet, data link distortions can upsetorthogonality and cause self-interference.

A transmitter may transmit a frequency reference “pilot” tone to thereceiver, for instance in a sub-band not used for conveying data. At thereceiver, a Phase Locked Loop (PLL) can be employed to synchronize thephase and frequency of the received signal to a receiver correlator suchthat only a small residual phase error exists in the PLL band. If thelink, including the analog conversion filters for the digital-to-analogand analog-to-digital converters in the transmitter and receiver,respectively, does not have linear response there will be changes inphase (i.e., dispersion) from band to band.

To address such offsets, cross-correlations for any set of wavelengthsmay be calculated by a filtering process. The multi-rate filter banks ofthe transmitter and receiver in one embodiment, as described above, havefilter coefficients that are wavelet functions, W(I,J). If there are Msub-bands in the filter then I ranges from 1 to M and J is the time ofthe current symbol at the transmitter; j is the time of that symbol whenit arrives at the receiver. For digital processing, the times aremeasured in samples, so both I and J are integers, and i is a specificsub-band to be processed in the receiver.

The modem sends symbols spaced in time by T for each of M samples sothat symbols are sent at . . . , J−3T, J−2T, J−T, J, J+T, J+2T, etc. Thesymbol sent at J arrives at time j in the receiver. Thus, a specificsymbol arrives at j+nT for processing in the receiver. The wavelets havelimited time duration and span only N symbols so the “n subscript” canbe limited to a small range of −(N-1) to +(N-1) to cover all thewavelets that overlap in time with the received symbol at time j.

The sub-bands are well defined in frequency so that any sub-band, i, hasonly a few overlapping neighbors. Thus, one need only consider bandsi+m, where m is in the range −2,−1,0,+1,+2 if, for example, there is nooverlap in frequency beyond the two nearest neighbor sub-bands on eitherside. That is, for magnitude m>2 this filter is in its stop-band. Ofcourse, i+m is always in the range of 1 to M, M being the total numberof sub-bands in the filter.

The transmitter encodes the bits in data symbols D(I,J) and sends anamplitude weighted wavelet for each sub-band. The symbols aretransmitted asSignal=sum {D(I,J)W(I,J)} over all I, all J.The sum of these terms for all active sub-bands arrives at the receiverat time j, where j=(J+delay in the link). Considering all except msub-bands are insignificant for receiving band iSignal for receiver band i at time j=sum {D(i+m,j+nT)W(i+m,j+nT)}.Thus, there is a sequence of wavelets overlapping in time andoverlapping in frequency when the sum is taken over all allowed m and n.

The receiver adjusts its timing and if necessary, the phase of thewavelet of the receiver filter so that the receiver filter evaluates theinner product with W(i,j) every symbol period, T, to obtainRecovered symbol=D(i,j)<W(i,j),W(i,j)>+(cross terms) for m, n≠0The notation <, > represents the inner product. A constant C₀₀represents the auto-correlation of the wavelet, i.e. the inner productwith itself, so that the data symbol is found by dividing by C₀₀. Thus,Normalized Recovered Symbol=D(i,j)+(cross terms)/C ₀₀where D(i,j) is the transmitted symbol D(I,J) delayed in time. Thewavelet functions are exactly, or nearly, orthogonal so that at thereceiver the cross term inner products are all zero or very smallcompared to C₀₀, by design. The receiver recovers the symbols accuratelyfrom the Normalized Recovered Symbol by threshold detection.

All the relevant wavelet correlations or inner products for band i canbe written as:<W(i+m,j+nT),W(k,j)>=C _(m,n)The subscripts range over plus and minus integers to represent theadjacent bands and adjacent symbol times, as previously defined.

However, dispersive impairments in the link may create a relative phaseshift between sub-bands. In that case, the “orthogonality” of thewavelets is lost and cross terms appear as self-interference in therecovered symbols.

In the presently disclosed invention, and as illustrated in FIG. 36,fixed tap-weight filters 540 are used to approximate the cross terms fora particular differential phase shift. The filter outputs are thenweighted 542 by a known function F(p) of the measured differential phaseshift and the weighted filter outputs are subtracted 544 from therecovered signal to remove the self-interference. In the preferreddesign, the fixed coefficients of the filter are calculated from theinner products at one value of differential phase shift. An adaptivecombiner could be used if the phases could not be measured.

In a preferred embodiment, the particular differential phase shift is 90degrees and the known function for weighting is sin(p), where p is themeasured differential phase angle.

If the phase of band i+m is angle Pm, then the neighboring bands havedifferential phase p=P₀−Pm. The inner products for a differential phaseangle, p, are first calculated to be:X _(m,n,p) =<W(i+m,j+nT,p),W(i,j)>

The self interference for band i has several components. For example,the recovered signal in band i is input to filter S. The recoveredsignal from the band to the left of i, which is band i−1, is insertedinto adjacent band filter A_(L) and the recovered signal from i+1 isinserted in the right filter A_(R) to the right of the band i. Theweight functions F(p) are determined from the p dependence of the Xcoefficients. The self-interference is then removed by combining withthe delayed recovered signal R, as in:Weighted sum=R−F ₁ *S−F ₂ *A _(L) −F ₂ *AR

The principle of calculating the self interference terms from the innerproducts of the wavelets can be extended from phase shift impairments toalso include time shift impairments by computing for a time shift t:Y _(m,n,t) =<W(i+m,j+nT,t),W(i,j)>The self interference due to timing offset is then removed by combiningfiltered signals in the same manner as for phase but with the filter tapweights chosen from the Y values.

Even if a sliding correlation can remove the timing offset betweentransmitter and receiver for one sub-band, there can be an offset inother bands due to group delay variation (i.e., dispersion).

The foregoing describes a method of providing phase and timeequalization for a wavelet modem based on cross-correlations of thewavelet functions. As noted, an adaptive combiner can be used ininstances where the phase or time shift cannot be readily measured.Further, since an m band modem has only m phase shifts relative to acorrelator reference phase, all m bands may be recovered with only thosem parameters. If the m angles can be measured, the combiner need not beadaptive. Yet, distortion within a wide sub-band or a frequency offsetmay complicate the calculation of cross-correlations, in which casemaking all taps adaptive may be mandatory.

In yet another application of inner-product cross terms, an isolatedwavelet W(i,j) with no overlap in time or frequency with other waveletsmay arrive offset in time from the receiver by u samples. The receivercomputes the isolated symbol as well as small early and late “satellite”responses at symbol times before and after the true symbol.Recovered Isolated Symbol=D(i,j)<W(i,j),W(i,j+u)>

The receiver can perform a sliding correlation as previously disclosedto find the correct time. However, the precision of the wide correlationpeak of the isolated signal for small errors of time, u, may not allowan exact determination of the setting where u=0. If, however, thecorrelation is also done at, for example, one symbol time before and onesymbol time after the true arrival of the peak then there will be twomore inner products:Early=D(i,j)<W(i,j),W(i,j+u+T)>Late=D(i,j)<W(i,j),W(i,j+u−T)>If the wavelets are symmetric or anti-symmetric, which is desirable fora linear phase response in a modem, then the Early and Late signals willbe equal in amplitude when the timing is correct, if the phase of thewavelets in the inner products match. That is, when u=0 the satelliteresponses from the receiver's filter have the same magnitude whensynchronized. Furthermore, the magnitude of the Early and Late signalsrelative to the Isolated signal can be used as a threshold to assurethat the sliding correlation is close to u=0 and not near a false peak.

For example, with one suitable wavelet function, the peak energy valuefor an isolated wavelet will be more than twice that of the energyoffset by one symbol time and will be more than sixteen times that ofthe energy offset by two symbol times. A suitable circuit for evaluatingsync detect is shown in FIG. 28 and is described subsequently.

In the absence of knowledge of the phase of the received signal relativeto the receiver's wavelet, then the receiver may also compute the crosscorrelations Early′ and Late′ from wavelet w(i,j), where w(i,j) isW(i,j) phase shifted by 90 degrees. In that case, the energy orspecifically the squared energies are:Early energy squared=(Early*Early)+(Early′*Early′)Late energy squared=(Late*Late)+(Late′*Late′)

The Early and Late energy or squared energy will be equal when u=0, thatis, when the transmitter and receiver are in time synchronization, sothat the incoming isolated wavelet aligns with the receiver's wavelet.Then, subsequent transmitted symbols can be detected by the receiverfilter. Furthermore, the ratio of squared energies for the isolated toearly or isolated to late signals are calculable from the crosscorrelations and that ratio can aid in the search for truesynchronization.

Small cross-correlations may also be present in the transmitted signalitself. These terms represent a slight non-orthogonality in the waveletswhich may be the result of the choice of a “near-perfect” wavelet familyand/or may be the result of truncations or similar efficiency-directedefforts in the transmitter hardware which degrade orthogonality. Theseeffects are also addressed in the receiver in a manner similar to thatfor addressing channel distortions.

For wavelets with high stop-band attenuation, the cross-correlations arecomputed as previously; namely,C(I,J)=<b,s|B,S>

-   -   for bands B=b±1 and symbols S=s±N

In practice, only the inter-symbol interference (ISI) for the sub-bandbeing equalized and its two nearest neighbors contribute. The threefilters have fixed taps and there is nothing requiring feedback. Thenormally small interference is removed by subtraction just as in thechannel-induced case described above.

The wavelet's imperfections are typically only noticed when transmittingextremely large numbers as would be the case for symbols carrying manybits at a correspondingly high bandwidth efficiency. The fixed filters,with tap weights proportional to the above equation, remove at thereceiver the self-interference that was generated inside thetransmitter.

In the following discussions, aspects of the modem receiver which enableequalization according to the foregoing requirements and synchronizationto the respective transmitter are discussed. FIG. 17 provides a blockdiagram of certain receiver circuitry which is invoked in thisdiscussion.

The previously discussed multiband equalizer uses the data in eachsub-band from the QMF filter bank to continuously track the phase andamplitude of each sub-band without the use of pilot tones. The modemreceiver uses this phase information both for data recovery by theequalizer and as the phase detector input to a phase-locked-loop usedfor transmitter timing synchronization.

FIG. 18 depicts an Equalizer (“EQ”) in a modem receiver capable ofperforming the functions described above. Functional blocks of theEqualizer 400 include a Filter 402 and a Quantizer 404 in the forwardpath and a Feedback circuit 406 connecting the Quantizer output to theFilter input. A Measuring circuit 408 is used to initialize the Feedbackcircuit. These circuits process each sub-band sequentially so allsub-bands are served by the illustrated circuits. This is possible sinceM sub-bands are transformed by the wavelet filter bank (labeled“Transform” in FIG. 17) 410 using M samples from the A/D converter 412.

The receiver multirate QMF wavelet filter bank 410 may be implementedwith a form of polyphase filtering followed by four M/2-point FFTs. Thisvery computationally efficient wavelet transform architecture isdescribed in the following. The FFT outputs successive sub-bands, oneper sample. As described, it is computed internally in a block-floatingpoint format with a mantissa and exponent that applies to all sub-bands;the FFT is essentially a fixed-point calculation but with extendedrange.

The separable wavelet functions described next are an example ofwavelets chosen for an efficient design implementation of the wavelettransforms for a modem.

A near-perfect reconstruction QMF filter bank can use FIR filters withcoefficients that are the product of a time-symmetric envelope function,E, multiplying a sinusoid. For the Master set of wavelets, which areused to transmit data, the sinusoids are preferred to be a cosine forodd numbered sub-bands and a sine for even numbered sub-bands. Thesymmetry of the functions is exploited to simplify the hardware.

Using the earlier notation, the wavelet for odd band, bo, and evenbands, be, at symbol time 0 is:|be, 0>=E*cos(w*t)

-   -   symmetric        |bo, 0>=E*sin(w*t)    -   anti-symmetric        The radian frequency is w, which is the frequency at the center        of each of M sub-bands.        w=[pi*(2*b−1)/(2*M)]    -   for band number b=[1:M]        Time, t, is measured in samples so the values of |b,0> are the        filter coefficients.

The length of the filter is L=M*N. To exploit the symmetries, N must bea multiple of 4 and is typically 8. M is typically a power of 2. Giventhese conditions, L is even and time t is:t=[−L/2+1/2:L/2−1/2]if L is odd t=[−L/2:L/2]

The envelope is chosen as the product of a sine function with a guassianfunction. The wavelet envelope is defined to be symmetric about themiddle of the filter by using the “fliplr” operator (flip columns leftto right about a vertical axis) in MATLAB (registered U.S. trademark ofThe MathWorks, Inc., Natick, Mass., USA) after defining the function forpositive time, tp.

-   -   a1=1.8244; preferred approximate    -   a2=2.2888; values    -   g=constant; determines the scaling in fixed point hardware        L=M*N;        p=3*M/2;        tp=[1/2:L/2−1/2];    -   for positive time if L is even        m=q2/p;        q1=pi*q2/(3*M);        q4=sin(a1*q1)./(a1*q1);        q5=exp(−m.*m/a2)·*q4;        E=g*[fliplr(q5),q5];

The preferred values of the constants a1 and a2 are chosen to minimizethe cross-correlations of the wavelets, thereby making the waveletsnearly orthogonal. For a modem, ‘near-orthogonality’ is usuallysufficient since there is always noise at the receiver. If theself-interference from the imperfect wavelets is too large, thecross-correlations can be computed and removed by a filter as alreadydescribed.

The filter bank can perform the filtering function by calculating allthe sums of products of the data with the Finite Impulse Response (FIR)coefficients. The sums of products of the sinusoids can be computed withFast Fourier Transforms (FFTs) over M samples.

Each wavelet has exactly 2*b−1 cycles of the sinusoid in M samples overeach of N segments. This symmetry leads to the definition of the odd andeven polyphase terms po and pe. The notation follows MATLAB conventions.po=s(1,:)−s(3,:)+s(5,:)−s(7,:);pe=s(2,:)−s(4,:)+s(6,:)−s(8,:);where s(i) are the samples from tap i out of the RAM shift registers ofthe receiver A/D buffer.

For symmetric wavelets, the filter output is:Master=SUM{po*cos( )−pe*sin( )}Slave=SUM{po*sin( )+pe*cos( )}

-   -   the sum is over M samples        For anti-symmetric:        Master=SUM{po*sin( )−pe*cos( )}        Slave=SUM{po*cos( )+pe*sin( )}

The arguments for the sinusoids are w*t and have radian frequency w:w=[(2*b−1)*pi/(2*M)] for band number b=[1:M]The wavelet time axis is offset by ½ from 0.

By comparison, the DFT for 4*M points is:X(k)=SUM{x(n)*W4^(n*k)}where W4=exp(−j*2*pi/(4*M))

So a 4M transform computes the right frequency, though there are onlyhave M points, the time axis needs to be shifted by ½, and only everyother sub-band is used from the Discrete Fourier Transform (DFT). All ofthese things are accounted for by the algebra given below.

The FFT multiplies by exp(−j*w*t) and integrates so if a complex signalis defined as:xs=complex(po,−pe); i.e. xs=po−j*pethen the FFT will compute:[po−jpe]*[cos−j sin]=[po*cos( )−pe*sin( )]−j*[po*sin( )+pe*cos( )]which is FFT(xs)=Master−j*Slave for symmetric wavelets.

Likewise, a complex signal for anti-symmetric wavelets is:xa=complex(pe,−po)which results in an FFT of the form:[pe*cos( )−po*sin( )]−j*[po*cos( )+pe*sin( )]=−[Master+j*Slave]

This means that a 4M transform can be used. In MATLAB, the M point inputis zero filled and, in the following, the unwanted terms are discarded.ys4=fft(xs,4*M);ys=ys4(2:4:2*M);

To complete the transform method, a time offset of ½ is needed and theconjugate for symmetric case puts the output in the form Master+j*Slave:kb=2*[1:2:M]−1;offset=exp(−j*kb*pi/(4*M));

-   -   % adds time offset of ½        y(1:2:M)=conj(ys.*offset);    -   % conj to make M&S positive        For the anti-symmetric case:        ya4=fft(xa,4*M);        ya=ya4(4:4:2*M);        kb=2*[2:2:M]−1;        offset=exp(−j*kb*pi/(4*M));        y(2:2:M)=ya.*offset;

The 4M transform contains many unneeded terms and can be simplified by adecimation in frequency to a 2M transform. For the symmetric case:xs=[xs, zeros (1,M)];mm=[0:2*M−1];W4=exp(−j*mm*2*pi/(4*M));

-   -   % complex twiddle        ys2=fft(xs.*W4,2*M);ys=ys2(1:2:M);    -   % 2M transform        y(1:2:M)=conj(ys.*offset);

The above 2M transform also has unwanted output terms and zero inputterms. It can be simplified to a 1M transform by another decimation infrequency.mm=[0:M−1];W4=exp(−j*mm*2*pi/(4*M));ys1=fft(xs.*W4,M);ys=ys1(1:M/2);

-   -   % M transform

Since there are only M/2 symmetric bands, a further simplification ispossible using a decimation in time. This leads to the final form forsymmetric and anti-symmetric processing with M/2 point transforms,namely:xf=xs.*W4;ys=(fft(xf(1:2:M),M/2)+W1.*fft(xf(2:2:M),M/2));y(1:2:M)=conj(ys.*offset(1:2:M));

-   -   % symmetric bands only

The above describes the receiver's wavelet transform. L samples from theA/D buffer are “collapsed” into M samples by the polyphase filtering,and then those M samples are processed by four M/2-point FFT's.

A simplified block diagram for the above algebra is found in FIG. 32. Inthat figure, the A/D output goes to a Shift Register of length L havingeight equally spaced taps which are weighted by the envelope in thePolyphase Filter, which in turn yields the signals po and pe arranged ascomplex numbers for input to the four FFTs. The M/2 points for the FFTsare sub-samples of the original stream of M samples per symbol so thateach of the complex multipliers and the adder shown operate at the samesample rate as the A/D. The complex multipliers use ROM tables ofexponentials given by the algebra above. The output goes to theequalizer and sync circuits.

Each M/2 point FFT requires a complex multiplier. The polyphase filterhas eight real taps and is roughly equivalent to one complex multiplier,so the overall circuit has the computational complexity of eight complexmultipliers per A/D sample.

With reference to FIG. 33, the transmitter uses the same QMF filter bankcoefficients following similar algebra but the data is first processedby M/2-point FFTs and placed in two RAM Shift Registers containingeither the sums for odd and even sub-bands, respectively, or containingthe sums for all bands in normal and time-reversed order, respectively.The output of the two RAM Shift Registers is then polyphase summed tocreate the transmitter signal for a D/A converter.

For the transmitter, the data bits for each sub-band are typicallygray-coded into symbols as numbers in the range +−1, +−3 . . . 2^n−1,etc. according to the number of bits per symbol, n. For the two RAMShift Registers, only the real or the imaginary part, respectively, areplaced into the Shift Registers so the output to the Polyphase Filter iseight real numbers per D/A sample.

The Filter circuit 402 of the receiver Equalizer 400 (FIG. 18) is animplementation of the previously disclosed approach to removingcross-correlations (i.e., loss of orthogonality) due to distortions inthe transmission medium (labeled “Data Link” in FIG. 18) or originatingin the transmitter itself, possibly as a result of using “near-perfect”wavelets. A practical modem may not have perfectly orthogonal waveletssuch that there are small, known cross-correlations even when thetransmitter and receiver are connected directly without distortions froma transmission medium.

The Quantizer 404 recovers the data symbols and provides as a feedbackmetric the difference between a quantized symbol (i.e., the Quantizeroutput) and the unquantized symbol (i.e., the Quantizer input).

The Measuring circuit 408 determines the approximate phase and amplitudeof isolated wavelets, which have been previously described as havingutility in modem synchronization. A Read Only Memory (ROM) table ofarctangents (not illustrated) within the Measuring circuit is used toestimate the phase and level of each active sub-band. Normally, thismeasurement is only made once, when a modem is first powered up on theoperating frequency. The preamble of isolated wavelets may repeat,however, to enable multiple modems to power up at any time.

In FIG. 19, components of the Feedback circuit 406 are illustrated inblock form. The Quantizer 404 error signal is received at a PhaseMeasurement Loop (“PML”) 430 and used as detailed below to generate aProportional Phase Correction. This correction signal is then utilizedwithin a Phase Measurement Unit (“PMU”) 418 in providing left, centerand right sub-band phase angles to the Equalizer Filter 402 for eachsub-band being processed.

Also shown in FIG. 19 is an Amplitude Measuring Loop (“AML”) 440 and anAmplitude Measuring Unit (“AMU”) 450. The Amplitude Measurement Loop 440uses the Quantizer Error signal from the Quantizer 404 in generating aGain Error Offset. This signal is then used within the AmplitudeMeasurement Unit 450 in generating a Gain Factor applied to EqualizerFilter outputs.

In the Phase Measuring Unit 418 of the Feedback circuit 406, illustratedin FIG. 20, the arctangent lookup value from the Measuring circuit 408is saved in Random Access Memory (“Arctan RAM”) 420 for eachsequentially analyzed sub-band. A Proportional Phase Correction in thecorrect direction (discussed below) increments a phase value stored in aPhase Angle RAM (“PAng. RAM”) 422 for the purpose of achieving phasetracking. The Phase Measurement Unit 418 computes the differential phaseangle between adjacent sub-bands since sequential processing makes theCenter Band Phase and the phases of the sub-bands to its left and rightaccessible with delays.

An important feature of the Phase Measurement Unit 418 is the Low PassFilter 424 located in the phase increment path. This filter providessmoothing independent of the update rate of the phase Feedback circuit406. The Low Pass Filter smoothes the band-to-band variations in thephase angle and tends to dramatically reduce the phase noise seen in theCenter Band Phase value. The signal into the Low Pass Filter isessentially the Center Band Phase. A process of copying, in reverseorder, the filter values at the beginning and end of the sequence ofactive sub-band phases removes end effects for the filter. The Low PassFilter circuit actually filters the differential phase betweensub-bands, then adds this filtered differential phase to the phase ofthe previous sub-band, stored in the PAng. RAM 422, to obtain the phaseof the current sub-band. RF Tuner phase noise is common across allsub-bands.

The Phase Measurement Loop 430 of FIG. 21 supplies the ProportionalPhase Correction signal utilized in the Phase Measurement Unit 418. In apractical modem, the gain of the Proportional Phase Correction isadjustable to provide one value for tracking the phase (e.g., a gainfactor of 2) and a different value for use in acquisition of the PhaseLocked Loop (e.g., a gain factor of ¼), which because it is notinitially locked can have a significant frequency error. The PhaseMeasurement Loop circuit computes the difference between two averages ofthe Quantizer Error signal using two Phase Measurement Filter circuits432, 434. Unlike Quadrature Amplitude Modulation (QAM), theconstellation is one-dimensional.

FIG. 22A illustrates four symbol amplitudes corresponding to two bitvalues transmitted in one sub-band. The horizontal arrows indicate theProportional Phase Correction from a nominal symbol amplitude of +3 toobserved symbol amplitudes on the right-most solid line. Symbolamplitude is plotted horizontally and is the amplitude of the Masterwavelet correlation after equalization. The vertical axis is the Slaveoutput and carries unnecessary redundant information. Each data symbol,at two bits per sub-band, is a point near one of the solid lines shown.A phase error exists such that the solid lines are not perfectly alignedwith the symbol values on the horizontal axis. The projection of thesymbol amplitude onto the horizontal axis determines the ProportionalPhase error. The standard deviation of the Proportional Phase Correctionis directly related to Bit Error Rate.

Because the modem uses Proportional Phase Correction, both the nominal+1 and +3 symbols can be treated identically. By symmetry, the −1, −3regions in the fourth quadrant are like those in the first quadrant, andthe symbols in the second quadrant are like those in the third quadrant.Thus, by employing the Phase Measurement Loop 430 circuitry toexclusive-OR 436 the sign of the Master and Slave signals out of theEqualizer Filter 402 and to appropriately modify the sign 438 of theresultant Proportional Phase Correction signal, all possible symbolvalues can be treated with a single diagram as shown in FIG. 22B.

The phase circuit does not depend upon an accurate measurement ofamplitude. The solid lines do not have to cross the horizontal axisexactly at +/−1 or +/−3 for the phase circuit to work correctly. Theaction of the phase feedback circuits causes the solid line to becomemore perpendicular to the horizontal Master data axis, while theamplitude feedback circuits (discussed subsequently) make the solid lineintersect the Master data axis at the Quantizer levels, which aremultiples of +/−1, +/−3, +/−5, etc.

The data points on the solid line above the horizontal axis can beaveraged together separately from the data points below the horizontalline. For that reason, the Phase Measurement Filters 432, 434 are calledTOP and BOT in FIG. 21. The difference, the Proportional PhaseCorrection, between TOP and BOT is approximately proportional to theangle of tilt in the solid line, which is the incremental phase errorfor this sub-band. As previously described, this error correction signalis selectively multiplied by a gain factor and subtracted from thestored phase angle value from RAM within the Phase Measurement Unit ofFIG. 20.

After collapsing the +/−3 and −1 symbols into a single +1 case, the tiltof the solid line can be removed even though the symbol amplitude is notexactly an integer. The phase is recovered independent of the amplitudeloop, though the two loops should have synchronized update rates so thetop and bottom averages are valid.

The Phase Measurement Filters 432, 434 average the Proportional PhaseCorrection signal by accumulating successive values and multiplying bythe reciprocal of the number of points accumulated. At high loopbandwidths, it may happen that the exclusive-OR 436 produces all top orbottom entries, in which case a logic signal (not shown) is definedwhich prohibits a change to the phase for that sub-band in the PhaseMeasurement Unit 418 for this update interval.

To account for the effect of gain error on observed quantizer error forhigher symbol levels, the Amplitude Measurement Loop 440 (FIG. 23)multiplies the Quantizer Error value by the reciprocal of the slicedsymbol value using a lookup ROM 442. The symbol value is provided by theQuantizer 404. The product is accumulated in a RAM 444 during an updateinterval and is provided as a Gain Error Offset. This offset value isthen input to an Amplitude Measurement Unit 450.

The amplitude of the recovered wavelets is measured at the same time asthe phase during initial power-up or when tuned to a new operatingfrequency range. The amplitude value is stored in RAM 452 of theAmplitude Measurement Unit 450, shown in block form in FIG. 24. TheAmplitude Measurement Unit 450 can also include a filter (not shown) forband-to-band smoothing in the update path similar to the Low Pass Filter424 in the Phase Measurement Unit 418. Such a filter in the AmplitudeMeasurement Unit 450 is used to reduce the fluctuations in gain due tonoisy feedback and thereby improves the Bit Error Rate.

The signal out of the Filter 402 is multiplied by the Gain Factor, whichis initially (i.e., at start-up) the reciprocal of the measuredamplitude of the isolated wavelet, divided by any scaling factor if notone. The Amplitude Measurement Unit 450 takes the isolated waveletamplitude directly from the FFT and uses a ROM 454 to provide thereciprocal scaled to represent a symbol of +1, a value which is referredto as “levelone.”

After this initial value of the reciprocal of levelone is stored as theGain Factor in RAM 452, the resulting input to the Quantizer 404 isapproximately normalized to the values +/−1, +/−3, +/−5, etc. times aconstant equal to the maximum magnitude of the Proportional PhaseCorrection. Subsequent updates are processed by the AmplitudeMeasurement Unit 450 on the basis of the Gain Error Offset from theAmplitude Measurement Loop 440. The Gain Factor is tracked according to:GF=GF×(1/(1+AverageGEO×FBGC))where GF=Gain Factor, AverageGEO=Average Gain Error Offset, andFBGC=Feedback Gain Constant which may have different values during loopconvergence and loop tracking. The quantity (1/(1+AverageGEO×FBGC)) isobtained in the Amplitude Measurement Unit 450 from a lookup ROM 456addressed by the Gain Error Offset.

The Quantizer 404 produces the Quantizer Error signal as the differencebetween the absolute value of an equalized signal and the signal roundedto the nearest odd integer. The Quantizer is based upon the followingMATLAB (registered U.S. trademark of The MathWorks, Inc., Natick, Mass.,USA) algorithm for quantizing a signal y. The Quantizer Error signal isreferred to as “minerr” in this algorithm. The recovered data symbol isreferred to as “slice.”

    x=abs(y);     x1=round(x);     slice=sign(y).*x1;     minerr=x−x1;    reven=find(2*fix.(x1/2)==x1);   if isempty(reven)==0    gt=find(x(reven)>=x1(reven));    slice(reven(gt))=sign(y(reven(gt))).*       (x1(reven(gt))+1);    minerr(reven(gt))=x(reven(gt))−       (x1(reven(gt))+1);    lt=find(x(reven)<x1(reven));    slice(reven(lt))=sign(y(reven(lt))).*       (x1(reven(lt))−1);    minerr(reven(lt))=x(reven(lt))−       (x1(reven(lt))−1); end

Thus, according to the foregoing, the master and slave QMF filter-bankoutputs are filtered with fixed tap equalization filters for correctingthe master and slave signals for various cross-correlation errorsappearing in the receiver. The fixed tap filter outputs are thenweighted and combined according to computer functions of the phaseangles to remove distortions and self-interference. Finally, thecombined equalizer filter outputs are trigonometrically weighted tocorrect for the overall phase shift in each sub-band due to channel orother distortions. Note that the order of steps in the algorithmdescribed above can be adjusted as one skilled in the art willappreciate.

Other particular features of the presently disclosed receiver Equalizerinclude the ability to combine data symbols into two groups, top andbottom, on the basis of a property of the symbols. For each group, anaverage quantizer error is calculated and used in tracking the phase ofeach sub-band. Differential phase variations between successivesub-bands are smoothed using a low-pass filter with band-edgecorrection.

The averaged quantizer error signal is also weighted by the inverse ofthe symbol value in the computation of a multiplicative gain factorapplied to the quantizer input, resulting in all symbols beingdistributed proximate the fixed, nominal symbol levels. Band-to-bandvariations in differential gain correction are smoothed by a low passfilter similar to that used for phase tracking.

The phase and amplitude loops in the receiver Equalizer are initializedwith values measured infrequently from isolated wavelets having knownamplitude and phase at the respective transmitter. These loops are thenupdated together or at least in synchronism.

A Phase-Locked-Loop in the receiver (not illustrated) serves tosynchronize the A/D sample rate to the transmitter D/A sample rate. Theanalog converter rates must be M times the symbol rate for a multirateQMF wavelet filter bank (except when the converter is over-sampling orunder-sampling).

FIG. 25 illustrates in block diagram form the circuitry utilized in thereceiver for providing timing and phase synchronization between thetransmitter and the receiver. The Preamble Detect 500 and Sync Predict502 functional blocks represent time domain synchronization circuitry.The Sync Detect 514 and Slide Balance 520 functional blocks representtransform or frequency domain synchronization circuitry. The componentsof this circuitry are addressed subsequently.

The D/A and A/D sample-rate clocks in the transmitter and receiver,respectively, must be phase locked. The Capture Frequency, FrequencyLock and Wavelet Discriminator of FIG. 25 provide this clocksynchronization, as is discussed in detail below.

Symbol sync is also a requirement. A wavelet extends over N symbols andeach symbol comprises M samples. Typical values include, but are notlimited to, M=256 and N=8. The transmitter and receiver must agree as towhen a wavelet starts. Another way of stating this is that the receivermust be able to identify which sample represents the start of a wavelet.The receiver refers to the difference between the transmitter andreceiver wavelet starting values as the “slide” (as used in a “slidingcorrelation”). The value of “slide” is thus a number from 0 to M−1.

If the preamble of isolated wavelets is such that multiple sub-bands arepulsed at once while still being isolated so that none overlap infrequency, then certain wavelet families will result in a set of verynarrow, high amplitude peaks of only a few samples in width.

A Preamble Detect circuit 500, shown schematically in FIG. 26, forms asignal referred to as “tap(i,j)” as a set of narrow pulses spaced M/2apart from such a preamble. The circuit determines the sample numberfrom 0 to M−1of each value of tap above a fixed or variable threshold.The sample number values are processed to find the left and right edgesof a cluster of peaks in the tap signal. The edges are then used in theSync Predict circuit 502 of FIG. 27 to compute the average center of thecluster. The average center value, also referred to as the averagelocation of the sample peak or “avglo” in FIG. 27, is the sample valuecoincident with the start of a wavelet as seen by the receiver samplecounter.

The transmitter starts the wavelet at a nominal sample count of zero.The timing offset of the transmit and receive sample counters isreferred to as “slide.” As illustrated in FIG. 27, the Sync Predictcircuit 502 actually has an ambiguity of +/−M/2 in its location of thewavelet and so two values of “slide,” labeled “sld1” and “sld2,” arecomputed.

An alternative method for synchronizing the receiver to the transmitterhas already been disclosed in the foregoing through the use of anisolated wavelet and a sliding correlation. One implementation of such aSync Detection circuit is shown in FIG. 28, and tests a center tapagainst first neighbor taps multiplied by 2 and second neighbor tapsmultiplied by 16. The tap weights depend upon the specific choice ofwavelet function. If all tests pass, the peak is in the center tap, thereceiver is synchronized, and the Sync Detect signal is output by theAND gate for the currently processed sub-band. The Majority Logic 512requires that more than half of all the sub-bands that issued anisolated wavelet at this symbol be detected.

In an illustrative hardware embodiment, M=256. Any number of sub-bandsmay be turned on, though 176 is a typical number of active sub-bands.The typical preamble thus consists of sending a symbol simultaneously onevery fourth active sub-band with no energy on the others. Then, afterN=8 symbol times on no energy, a second group of every fourth symbol issent, and so on. The preamble is thus composed of seven null symbols, aknown symbol value in active sub-bands, seven null symbols, a knownsymbol value in active sub-bands, etc.

If one of the “sld1” or “sld2” signals from the Sync Predict circuit ofFIG. 27 is incorrectly offset by M/2, there will be no coincident SyncDetect output from the Sync Detect circuit 514 of FIG. 28; thischaracteristic can be used to correctly resolve the M/2 ambiguity of theSync Predict circuit 502.

The Sync Predict circuit of FIG. 27 finds candidate “slide” valueswithout the aid of the receiver QMF filter bank, whereas the one symboltime offset tap values from the Sync Detect logic of FIG. 28 are used inthe Slide Balance circuit 520 of FIG. 29; the sliding taps in the ShiftRegister of the Band-Shared Section (see FIG. 17) are then adjusted tothe presumed start of a wavelet by trial and error in minimizing theslide balance. When synchronized, the early and late squared energiesare equal. When not synchronized, the two signals are not balanced; thatdifference indicates not only that the slide value is incorrect but alsoindicates which is the proper direction to make the slide value correct(or more correct).

The “slide” value synchronizes the start of the symbols by requiring thebuffers for the A/D input shift register connected to the transform tobe moved (or the taps to be moved for addressing the RAM). Thus, thetaps are slid by the amount of the computed “slide” which is output fromthe Slide Balance circuit 520 of FIG. 29. This requires considerabledelay to flush all the calculations in the wavelet filters.

When the balance reaches approximately zero, the counter, mctr, whichdefines the start of wavelets and also the order of the sub-bandprocessing, is in step with the corresponding mcntr in the transmitter.The initial value in mctrw changes when power is applied to the circuitand slide is added to that value to achieve symbol sync. The advance andchange signals from FIG. 29 go into the state-machine that control allthe logic in the receiver.

While a state-machine for the receiver is not illustrated explicitly, itwill be readily apparent to those skilled in the art that some form ofcontrol mechanism is necessary for controlling logic elements disclosedin the accompanying drawings.

The preamble of groups of time and frequency isolated wavelets has adistinctive property for the wavelets selected. They appear as animpulse of only a few samples wide even though the wavelet is 2048samples long (M×N). The preamble appears as a set of several narrowspikes spaced M/2 apart, each spike being five to seven samples wide;everywhere else, the time domain signal is nearly zero.

The spikes are biggest at the center of the wavelet's duration. The A/D412 (FIG. 17) input to the transform 410 already has eight taps spaced Msamples apart. Those taps are used as the input to the Preamble Detectcircuit 500 of FIG. 26. The sum of the absolute values results in thecomb-like pattern of pulses all about the same height, five to sevensamples wide, and spaced M/2 apart. The problem is to find the averagecenter of these pulses, which defines the exact center of the waveletenergy. Because of the M/2 spacing, the slide value or the slide valueplus M/2 modulo M is provided; these two values are referred to in FIG.27 as sld1 and sld2.

Unlike the frequency-domain circuit which finds slide by trial and errorin a sliding correlation, the circuit in FIGS. 26 and 27 can find theslide value before the wavelet has even reached its center owing to thespikes spaced every M/2 apart. In practice, the M/2 ambiguity isresolved by the receiver's state machine (not illustrated) by observingthat the frequency domain Sync Detect of FIG. 28 only has an output whensync is correct to within less than M/2. The logic is thus capable ofpicking the correct value between sld1 and sld2.

The bottom half of FIG. 26 is concerned with finding the center of thespikes, which as described are approximately six samples wide. Asdiscussed above, the mctrw value cycles from zero to M−1. Whenever thespike is above a noise threshold, the counter values are saved in theshift register. If the values are in a contiguous, ascending order, thenthe inside of a spike has been identified. Otherwise, this discontinuityis a left edge (edgeL) or right edge (edgeR) which will be used as acontrol signal, identified in FIG. 27 by dotted lines. The controlsignals are used to latch the mcntC value of the left and right edgesinto the registers rLL and rRR. The counter values are then used to getthe count for the center of the spike. All of the centers from all ofthe spikes are averaged over n spikes to arrive at the slide values sld1and sld2, typically accurate to +/− one sample.

Note that the Preamble Detect and Sync Predict circuits 500, 502 canonly measure “slide” to within one sample, whereas the balance techniqueassociated with the Sync Detect and Slide Balance circuits 514, 520 canmeasure “slide” to a small fraction of a sample. The measurement ofarrival times of the wavelet (i.e., symbol) is also necessary tosynchronize a multitude of transmitters in a multiple access system ofthe type previously disclosed.

The modem must also synchronize the sample clock using a PLL. To acquiresample rate lock, a coarse frequency correction can be made based uponthe known interval between transmission of the isolated wavelet insuccessive preambles. The coarse frequency is corrected if the receiverdoes not find the known interval. To aid in this, the Slide Balancecircuit provides a Preamble Gate (PG) signal, the latter beingindicative of when data has stopped and the preamble is present. Theright edge control signal, edgeR, is ANDed with PG. Data after thepreamble does not have energy gaps and hence no left edges. At poornoise levels, there is a need to make the PG signal robust. It is passedthrough the logic equivalent of a monostable multivibrator to stretchthe pulses (FIG. 27) so they overlap and thereby produce a clean pulseout of PG for the duration of the preamble.

The time, in samples, between Preamble Gates minus the known timeinterval in the transmitter is the coarse input to a VCXO.Alternatively, a more stable coarse signal can be obtained from thechange in the slide value over one or more preambles; four preambles areused in the circuit of FIG. 30 to generate a signal proportional to thecoarse frequency error. This signal may have an M/2 ambiguity.

When the receiver power is first applied, the receiver must adjust itsoscillator (a VCXO) to have the same frequency as the transmitter.Initially, however, the frequency offset may be so large that thewavelet engine cannot be used. In this case, the interval between thepreambles is used as a coarse frequency reference.

The problem for coarse frequency acquisition is to accurately detect thearrival time of the isolated wavelets in the preamble. This is exactlywhat the time-domain Sync Predict circuit does, so the sld1 and sld2outputs are used as the input to Capture Freq logic of FIG. 30. Usingexemplary numbers, if the preambles are spaced at 2048 symbols apart,then a counter could count 2049 if the local clock is slightly fast or2040 if very slow. For more accuracy, the measurement can be taken overa longer time. A change in the computed slide over four preambles isused in this logic to generate a signal proportional to the coarsefrequency error.

Once the coarse error from the preamble is applied, the VCXO presumablyslews to nearly the correct frequency, a condition of frequency lock. Toachieve phase lock, the slide must then be applied to move the A/Dbuffer readout timing and the Equalizer must start to converge on thecorrect phase by the action of the Phase Measurement Loop and PhaseMeasurement Unit disclosed above.

A D/A (not shown) supplies control voltage to a conventional VCXO. Thechanges in phase following the initial measurement of a selectedsub-band (or some average of sub-bands especially with the Low PassFilter 424 present) are then used as the fine correction to the VCXO. InFIG. 30, the Capture Freq circuit connects the most significant bits ofa D/A to the coarse frequency correction of the VCXO. The lesssignificant bits from the D/A are connected to the phase correction andare derived from the Freq Lock circuit of FIG. 31. The wavelet receiverthen acts as a phase detector. The output from the Phase Meas Loop 430(FIG. 21) is latched in the Freq Lock circuit of FIG. 31 for a specificsub-band (e.g. B1 or B2). The VCXO is then locked to the selectedsub-band. The modem receiver thus uses a wavelet phase detector toachieve phase lock after first getting frequency lock using the intervaltime for isolated wavelets in a preamble.

The phase detector output is combined with the coarse slide differencesignal to achieve the two step acquisition. If the equalizer cannottrack phase when the frequency offset is determined by the coarsecircuit alone, then a third step may be needed between the first two.This third step uses the wavelet transform but not the equalizer. Itrelies on the fact that the transform is a filter bank and when thefrequency of the receiver is wrong, the sub-bands of the filter bankwill be shifted in frequency so that energy from the isolated waveletsin the preamble will leak unequally into the adjacent sub-bands.

This is analogous to a crude frequency discriminator and is identifiedin FIG. 25 as a Wavelet Discriminator (labeled “W Discriminator” in thefigure). This brings the VCXO closer to the correct frequency andthereby allows the equalizer to track phase. In practice, however, stepthree is usually not needed because the equalizer is capable of trackinga large frequency offset.

During the transition from coarse to fine control of the VCXO, theEqualizer must be able to track a considerable frequency offset(typically ½ part per million) which may require a larger feedback gainthan is normally used after phase lock is attained. Thus, a selectablegain factor, not illustrated, is preferably applied to the ProportionalPhase Correction input to the Phase Measurement Unit 418 of FIG. 20.

To reiterate, significant features of the timing circuits discussedabove include achieving coarse frequency lock of the sample rate byapplying to a VCXO (or VCO) the difference, in samples, of the intervalbetween isolated wavelets as measured in the receiver and the knowninterval at the transmitter. In the foregoing, this difference isreferred to as the variable “slide” (or “sld1” or “sld2”). An additionalsignificant feature of the presently described modem receiver is theability to achieve phase lock by applying the phase from the receivedwavelets to the VCXO. The coarse and fine signals may overlap in theirrange of control voltage to the VCXO.

As discussed, the present modem receiver provides the ability to performpreamble detection using the sum of absolute values of A/D samples fromequally spaced taps. Cluster processing of sample values for selectedvalues of “tap” is utilized to determine the average position of thewavelet in time as an offset, referred to as the variable “slide,”between the transmit and receive sample counters. Multiple groups ofisolated wavelets are sent as a cluster in the preamble and the meanlocations of the groups are averaged. Such processing finds the edges ofthe cluster in sample count values and determines the mean value as thelocation of the wavelet.

Another issue which the presently disclosed modem addresses aremultipath echoes which, for multiple sub-bands, are a problem for highspeed communications. A method of dealing with this problem in thepreviously described modem receiver is outlined below. Small randomphasing effects like Doppler shift are ignored.

In the FIG. 34, the horizontal phasor, B, represents the transmittedsignal phase for all the sub-bands in a multiband system. They all havethe same phase at the transmitter. Here they are shown arriving at thereceiver at the same phase assuming that there is no other impairmentexcept for multipath. The receiver has rotated B to zero degrees.

Phasor A is the multipath reflection that arrives at the receiver for aspecific sub-band. Each sub-band will have a particular phase angle sothat the tips of all these trace out a circle. If the sub-bands b arenumbered, so 1≦b≦M, and the echo delay, measured in samples, is d, thenthe phase angle P of each echo in a multiband (e.g. wavelet) modem is:P=(2*b−1)*d*90/(M+e)measured in degrees where e is some constant phase, possibly 0, that iscommon to all and is attributable to the reflection process. The lengthA of the echo phasors will be a fraction, a, times the length of BA=a*B with 0<a<1The vector sum resultant phasor, C, is what is actually measured by thereceiver. Clearly, the amplitude of C has a quasi-sinusoidal amplitudeand phase across all the sub-bands.

When the phase of A is a multiple of 180 degrees then there isdestructive interference (fading) in that sub-band and the Signal toNoise Ratio (SNR) suffers, so the Bit Error Rate (BER) is worse. Whenthe phase is a multiple of 90 or 270 degrees then the modem is notaligned to the true signal for those sub-bands and there is Inter-SymbolInterference (ISI) which affects the BER. The wavelet equalizer can usethe second derivative of the phase with respect to sub-band frequency toweight the ISI correction for phase error, which is related to groupdelay or arrival time error, since the ISI is largest when the angle ofA passes through peaks near 90 and 270 degrees. While this wasdetermined experimentally to some effects of ISI, it does not help theSNR fading loss that occurs at 180 degrees.

The following discusses a way to track the true signal B rather than theresultant C, thereby eliminating fading. The technique has applicationfor both wireless and wireline (including CATV, powerline, phoneline,etc.) communications.

It is assumed that the modem, at initialization, has measured the phaseand amplitude of C for each active sub-band out of the total Msub-bands. From these measurements, the three parameters a, d, e can befound since they are common for all sub-bands. As a result of a fit tothe data, through a feedback circuit or simple approximations, a isfound from the max/min amplitude; d is found from the frequency of thephase ripple or from zero-crossings of phase ripple across thesub-bands; and e is an additive constant common to all sub-bands formaking the phase curve fit the formula given above. For threeparameters, more than 3 sub-bands are required if the fading is not flatacross frequency.

The measured phase angles of the active sub-bands, Pang, is related tothe phase, P, of vector A by the geometry in FIG. 34. P is obtained froma, d, and e by the formula presented above and noting:Tangent(Pang)=A*sin(P)/(B+A*cos(P))and that the length of the true vector B is:B=C*cos(Pang)−A*cos(P)The signal C contains ISI due to vector A that must be subtracted out ofthe previous calculation. The amount of ISI depends on delay orequivalently the derivative of the angle P with respect to the sub-bandand it depends on the length of A (=a*B). The removal of ISI for awavelet modem follows the prescription given above and summarized here:Quantizer input=B=C*cos(Pang)−A*cos(P)−ISI

To track the vector B in real time, one method using a circuit such asshown in FIG. 35 is to first initialize to the correct value of B usingthe formulas just presented. Then, the tracking loop converges to followthe angle of B (which is shown in FIG. 34 as zero for all sub-bands butwould in fact have variations due to the transmission link). Theamplitude of the wavelet transform is then be C*cos(Pang) rather than C.The transform is projected in the direction of the true signal. So, toobtain B+ISI, the modem feeds back the quantizer input B and subtractsa*B*cos(P) from the projection C*cos(Pang). The modem then computes andsubtracts the ISI term prior to input to the quantizer. The quantizererror fed back to the phase and amplitude tracking circuits maintainsthe tracking of phasor B (rather than C). The quantizer input will notexhibit fading.

The modem can determine all three vectors C, B, and A. Just as theinterference due to A was subtracted out to get B, it is also possibleto have another, similar circuit solve for A by subtracting out the ISIdue to B with a separate phase tracking loop. These circuits can trackeach sub-band coherently for both A and B. Thus the vector's magnitudesB and A=a*B can then be added together to get constructiveinterference—a bigger signal by a factor of 1+a than the ‘direct path’signal B. Obviously, this processing is not worthwhile for small a.

The technique described above uses an equalizer in a multiband modem toeliminate the destructive interference of multipath communicationsassuming a single echo path. It also proposes converting the destructiveinterference into constructive interference so that the SNR and BER maybe better than without the echo. It applies to any multiband modemalthough the wavelet modem of the specific examples is used for claritybecause the wavelet modem equalizer easily removes ISI and tracks phase.

If there are multiple reflection paths at different distances, thusresulting in different delays, then the phase versus sub-bandinformation from the phase tracking loops of the modem can be bandpassfiltered to distinguish the phase ripple frequency associated with eachdelay as implied in the formula (2*b−1)*d*90/M. The multiple trackingcircuits for each echo path would then be analogous to the multiple armsof a spread-spectrum Rake Receiver. Rather than bandwidth-wastingspreading, a form of redundancy, the method disclosed herein getsdiversity through multiple sub-bands to solve the fading problem at fargreater throughput.

The modem angle defined in the hardware as Pang is actually the anglebetween the transmitter and the receiver signal being tracked; in theabove Pang is the angle between B and C. Thus, the signal has alreadybeen rotated to eliminate the transmitter-to-receiver phase shift ineach sub-band.

When there are multiple reflections along paths, i, at angles, Pi,relative to B, and each reflection has a length B*Ai, then with the sumstaken over i (“SUM”):C*cos(Pang)=B*[1+SUM{Ai*cos(Pi)}]C*sin(Pang)=B*SUM{Ai*sin(Pi)}The SUMs will be different for each sub-band but they can be trackingparameters:C*cos(Pang)=B*(1+Ui) for Ui=SUM{Ai*cos(Pi)}, −1<Ui<+1C*sin(Pang)=B*Vi and Vi=SUM{Ai*sin(Pi)}, −1<Vi<+1By squaring and adding and solving for B:B=C/sqrt[Vi^2+(1+Ui)^2]−Ki*ISIThe denominator term can be approximated so:B=C/(1+Hi)−Ki*ISI

The phase in degrees of each multipath A varies as di*(b−1)*90/M forpath delay di. The group delay, which is the derivative with respect toband number b, is proportional to di and so all paths can be additivelycombined into Ki*ISI where ISI for symbol n is found from the ISI atsome reference delay such as t=M/4.

The time shift impairment for group delay was defined in the foregoingas:Ym,n,t=<W(i+m,j+n*T,t),W(i,j)>A modem equalizer would normally have this filter Ym,n,t available tocompensate for both positive and negative group delay. For multipath,only the positive delay ISI filter is be multiplied by Ki to compensatefor the ISI due to Ai in the above formula for B.

To control the required non-linear feedback loop, a preferredarrangement is to have a reference signal path with a referencequantizer and one or more trial paths with trial quantizers. Eachquantizer requires a gain parameter Gi ahead of the quantizer to set thelevels to odd integers. Starting with initial values corresponding to nomultipath as a reference, the trial paths increase or decrease thevalues Ui, Vi, Ki and the difference in quantization error from the 3respective paths determines the size and direction of a modification tothe parameters. If a trial yields a better (i.e., lower) quantizationerror than the reference then those parameters are used as the nextreference; otherwise the trials continue using the previous reference.

Having described preferred embodiments of the invention, it will nowbecome apparent to one of ordinary skill in the art that otherembodiments incorporating the concepts may be used. It is felt,therefore, that these embodiments should not be limited to disclosedembodiments but rather should be limited only by the spirit and scope ofthe appended claims.

1. A multi-band equalizer for a modem receiver, comprising: aself-interference canceling filter for receiving an input signal, forremoving from the input signal cross correlation terms resulting frominput signal self-interference and for providing a corrected inputsignal, the self-interference canceling filter comprising a fixedtap-weight filter bank for filtering the input signal, a weightingcircuit for weighting the filtered input signal with a pre-determinedfunction of a differential phase angle between adjacent subbands of theinput signal, and a subtraction circuit for subtracting the weighted,filtered input signal from the input signal to generate the correctedinput signal; a quantizer for quantizing the corrected input signal andfor providing a quantizer error signal based upon a difference betweenthe corrected input signal and the quantized, corrected input signal;and a feedback circuit for calculating a proportional phase correctionmetric from the quantizer error signal and for providing phase valuesfor plural subbands, derived from the proportional phase correctionmetric, to the self-interference canceling filter.
 2. The multi-bandequalizer of claim 1, wherein the feedback circuit is operative tocalculate the differential phase angle from the proportional phasecorrection metric.
 3. The multi-band equalizer of claim 1, wherein thefeedback circuit comprises a phase measurement loop for calculating thea difference between values of the quantizer error signal as theproportional phase correction metric.
 4. The multi-band equalizer ofclaim 3, wherein the values of the quantizer error signal are averagevalues each based upon plural input signal values.
 5. The multi-bandequalizer of claim 1, wherein the feedback circuit comprises a phasemeasurement unit operative to calculate the differential phase anglebetween adjacent subbands of the input signal from the proportionalphase correction metric.
 6. The multi-band equalizer of claim 5, whereinthe phase measurement unit comprises a data storage element for storingdata representative of a phase value for each subband of the inputsignal and logic for updating the phase value for each subband on thebasis of the proportional phase correction metric.
 7. The multi-bandequalizer of claim 5, wherein the phase measurement unit comprises a lowpass filter for smoothing the proportional phase correction values forsuccessive subbands.
 8. The multi-band equalizer of claim 5, wherein thephase measurement unit further comprises temporal delay circuits wherebythe phase measurement unit is adapted to provide the phase values forplural subbands at a given time.
 9. The multi-band equalizer of claim 1,further comprising a measurement circuit for determining approximatevalues of phase and amplitude of isolated data units of the input signaland providing the approximate values to the feedback circuit as initialreference values for calculation of the proportional phase correctionmetric prior to receipt of the quantizer error signal.
 10. Themulti-band equalizer of claim 1, further comprising an amplitudemeasurement loop for multiplying the quantizer error signal by a valueinversely proportional to the quantized, corrected input signal andproviding a product of the multiplication as a gain error offset value.11. The multi-band equalizer of claim 10, wherein the amplitudemeasurement loop further comprises a data storage element foraccumulating successive multiplicative products of the quantizer errorsignal and the value inversely proportional to the quantized, correctedinput signal and providing the accumulated successive multiplicativeproducts at a predetermined temporal interval as the gain error offsetvalue.
 12. The multi-band equalizer of claim 10, wherein the valueinversely proportional to the quantized, corrected input signal is theinverse of the quantized, corrected input signal.
 13. The multi-bandequalizer of claim 10, further comprising an amplitude measurement unitfor receiving the gain error offset value from the amplitude measurementloop and for deriving a gain factor therefrom.
 14. The multi-bandequalizer of claim 13, further comprising a multiplier for multiplyingthe corrected input signal by the gain factor prior to quantization bythe quantizer.
 15. The multi-band equalizer of claim 13, wherein theamplitude measurement unit is operative to calculate a new value for thegain factor based upon a multiplicative product of a previous value ofthe gain factor with the inverse of one plus the product of the gainerror offset times a feedback gain constant.
 16. The multi-bandequalizer of claim 15, wherein the value of the inverse of one plus theproduct of the gain error times a feedback gain constant is stored in adata storage element in the amplitude measurement unit and is indexed bythe gain error offset from the amplitude measurement loop.
 17. Themulti-band equalizer of claim 15, wherein the amplitude measurement unitfurther comprises a low-pass filter for smoothing successive new valuesfor the gain factor.
 18. The multi-band equalizer of claim 15, whereinthe amplitude measurement unit further comprises a data storage elementfor storing gain scale factors, whereby the amplitude measurement unitreceives amplitude data for isolated data units of the input signal andindexes the data storage element with the amplitude data to obtain again scale factor.
 19. A multi-band equalizer for a modem receiver,comprising: a quantizer for receiving an input signal, for quantizingthe input signal, and for providing a quantizer error signal based uponthe difference between the input signal and the quantized input signal;and an amplitude measurement loop for multiplying the quantizer errorsignal by a value inversely proportional to the quantized input signaland for providing a product of the multiplication as a gain error offsetvalue; an amplitude measurement unit for receiving the gain error offsetvalue from the amplitude measurement loop and for deriving a gain factortherefrom; and a multiplier for multiplying the input signal by the gainfactor prior to quantization of the input signal by the quantizer. 20.The multi-band equalizer of claim 19, wherein the amplitude measurementunit is operative to calculate a new value for the gain factor basedupon a multiplicative product of a previous value of the gain factorwith the inverse of one plus the product of the gain error offset valuetimes a feedback gain constant.
 21. The multi-band equalizer of claim20, wherein the value of the inverse of one plus the product of the gainerror offset value times a feedback gain constant is stored in a datastorage element in the amplitude measurement unit and is indexed by thegain error offset value from the amplitude measurement loop.
 22. Themulti-band equalizer of claim 20, wherein the amplitude measurement unitfurther comprises a low-pass filter for smoothing successive new valuesof the gain factor.
 23. The multi-band equalizer of claim 20, whereinthe amplitude measurement unit further comprises a data storage elementfor storing gain scale factors, whereby the amplitude measurement unitreceives amplitude data for isolated data units of the input signal andindexes the data storage element with the amplitude data to obtain again scale factor.
 24. The multi-band equalizer of claim 19, furthercomprising: a self-interference canceling filter for receiving the inputsignal and for providing a corrected input signal as the input signalinto the quantizer; and a feedback circuit for calculating aproportional phase correction metric from the quantizer error signal andfor providing phase values for plural subbands, derived from theproportional phase correction metric, to the self-interference cancelingfilter.
 25. The multi-band equalizer of claim 24, wherein theself-interference canceling filter comprises: a fixed tap-weight filterbank for filtering the input signal; a weighting circuit for weightingthe filtered input signal with a pre-determined function of adifferential phase angle between adjacent subbands of the input signal;and a subtraction circuit for subtracting the weighted, filtered inputsignal from the input signal to generate the corrected input signal. 26.The multi-band equalizer of claim 25, wherein the feedback circuit isoperative to calculate the differential phase angle from theproportional phase correction metric.
 27. The multi-band equalizer ofclaim 24, wherein the feedback circuit comprises a phase measurementloop for calculating a difference between averaged positive values ofthe quantizer error signal and averaged negative values of the quantizererror signal as the proportional phase correction metric.
 28. Themulti-band equalizer of claim 24, wherein the feedback circuit comprisesa phase measurement unit operative to calculate the differential phaseangle between adjacent subbands of the input signal from theproportional phase correction metric.
 29. The multi-band equalizer ofclaim 28, wherein the phase measurement unit comprises a data storageelement for storing data representative of a phase value for eachsubband of the input signal and logic for updating the phase value foreach subband on the basis of the proportional phase correction metric.30. The multi-band equalizer of claim 28, wherein the phase measurementunit comprises a low pass filter for smoothing the proportional phasecorrection values for successive subbands.
 31. The multi-band equalizerof claim 28, wherein the phase measurement unit further comprisestemporal delay circuits whereby the phase measurement unit is adapted toprovide the phase values for plural subbands at a given time.
 32. Themulti-band equalizer of claim 24, further comprising a measurementcircuit for determining approximate values of phase and amplitude ofisolated data units of the input signal and providing the approximatevalues to the feedback circuit as initial reference values for thecalculation of the proportional phase correction metric prior to receiptof the quantizer error signal.
 33. A method of processing input data ina multi-band equalizer to realize recovered data, comprising: filteringthe input data in a filter to remove self-interference cross correlationterms from the input signal resulting from self-interference and toprovide phase and time delay corrected input data by filtering the inputdata using a fixed tap-weight filter bank, weighting the filtered datausing a predetermined function of a differential phase angle betweenadjacent subbands of the input data, and subtracting the weighted,filtered input data from the input data to generate the phase and timedelay corrected input data; quantizing the corrected input data by aquantizer and providing a quantizer error signal as the differencebetween the corrected input data and the quantized, corrected inputdata; and calculating a proportional phase correction metric from thequantizer error signal and providing phase values for plural subbands,derived from the proportional phase correction metric, to the filter foruse in removing the self-interference in the input data.
 34. The methodof claim 33, wherein the phase and time delay corrected input dataprovided by the step of filtering comprises master and slave correlationvalues.
 35. The method of claim 34, wherein the step of calculatingfurther comprises dividing quantizer error signal values according to asign of the slave correlation values and calculating a differencebetween averaged values of the quantizer error signal for positive slavecorrelation values and averaged values of the quantizer error signal fornegative slave correlation values as the proportional phase correctionmetric.
 36. The method of claim 35, wherein the step of calculatingfurther comprises calculating the differential phase angle betweenadjacent subbands of the input data from the proportional phasecorrection metric.
 37. The method of claim 33, further comprising thestep of low-pass filtering the proportional phase correction metric forsuccessive subbands.
 38. The method of claim 33, further comprising:estimating the phase and amplitude of isolated data units of the inputdata and using the estimated phase and amplitude as initial referencevalues for the calculation of the proportional phase correction metricprior to the calculation of the quantizer error signal.
 39. The methodof claim 33, further comprising: multiplying the quantizer error signalby a value inversely proportional to the quantized, corrected input datato realize a gain error offset value.
 40. The method of claim 39,further comprising: deriving a gain factor from the gain error offsetvalue and multiplying the phase and time corrected input data by thegain factor prior to the step of quantizing.
 41. The method of claim 40,further comprising: low pass filtering successive values of the gainfactor prior to multiplying the phase and time corrected input data bythe low-pass filtered gain factor.